Open Access

A result on three solutions theorem and its application to p-Laplacian systems with singular weights

Boundary Value Problems20122012:63

DOI: 10.1186/1687-2770-2012-63

Received: 16 February 2012

Accepted: 18 May 2012

Published: 22 June 2012

Abstract

In this paper, we consider p-Laplacian systems with singular weights. Exploiting Amann type three solutions theorem for a singular system, we prove the existence, nonexistence, and multiplicity of positive solutions when nonlinear terms have a combined sublinear effect at ∞.

MSC:35J55, 34B18.

Keywords

p-Laplacian system singular weight upper solution lower solution three solutions theorem

1 Introduction

In this paper, we study one-dimensional p-Laplacian system with singular weights of the form where φ p ( u ) = | u | p 2 u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq1_HTML.gif, λ is a nonnegative parameter, h i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq2_HTML.gif, i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq3_HTML.gif is a nonnegative measurable function on ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq4_HTML.gif, h i 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq5_HTML.gif on any open subinterval in ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq4_HTML.gif and f , g C ( R + , R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq6_HTML.gif with R + = [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq7_HTML.gif. In particular, h i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq2_HTML.gif may be singular at the boundary or may not be in L 1 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq8_HTML.gif. It is easy to see that if h i L 1 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq9_HTML.gif, then all solutions of ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif) are in C 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq11_HTML.gif. On the other hand, if h i L 1 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq12_HTML.gif, then this regularity of solutions is not true in general; for example, even for scalar case, if we take h ( t ) = ( p 1 ) t 1 | 1 + ln t | p 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq13_HTML.gif, p > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq14_HTML.gif and λ = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq15_HTML.gif, f 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq16_HTML.gif, then h L 1 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq17_HTML.gif, and the solution u for corresponding scalar problem of ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif) is given by u ( t ) = t ln t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq18_HTML.gif which is not in C 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq11_HTML.gif.
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equa_HTML.gif
For more precise description, let us introduce the following two classes of weights;
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equb_HTML.gif

We note that h given in the above example satisfies h A B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq19_HTML.gif but h L 1 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq20_HTML.gif. The main interest of this paper is to establish Amann type three solutions theorem [4] when h i A B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq21_HTML.gif with possibility of h L 1 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq20_HTML.gif. The theorem generally describes that two pairs of lower and upper solutions with an ordering condition imply the existence of three solutions. Recently, Ben Naoum and De Coster [6] have proved the theorem for scalar one-dimensional p-Laplacian problems with L 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq22_HTML.gif-Caratheodory condition which corresponds to case h L 1 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq23_HTML.gif; Henderson and Thompson [18] as well as Lü, O’Regan, and Agarwal [23] - for scalar second order ODEs and one-dimensional p-Laplacian with the derivative-dependent nonlinearity respectively; and De Coster and Nicaise [11] - for semilinear elliptic problems in nonsmooth domains. For noncooperative elliptic systems ( p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq24_HTML.gif) with k i 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq25_HTML.gif and Ω bounded, one may refer to Ali, Shivaji, and Ramaswamy [3]. Specially, for subsuper solutions which are not completely ordered, this type of three solutions result was studied in [26].

The three solutions theorem for our system ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif) or even for corresponding scalar p-Laplacian problems is not obviously extended from previous works mainly by the possibility h L 1 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq20_HTML.gif. Caused by the delicacy of Leray-Schauder degree computation, the crucial step for the proof is to guarantee C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq26_HTML.gif regularity of solutions, but with condition h A B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq27_HTML.gif, C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq26_HTML.gif regularity is not known yet. Due to the singularity of weights on the boundary, the C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq26_HTML.gif regularity heavily depends on the shape of nonlinear terms f and g. Therefore, the first step is to investigate certain conditions on f and g to guarantee C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq26_HTML.gif regularity of solutions. Another difficulty is to show that a corresponding integral operator is bounded on the set of functions between upper and lower solutions in C 0 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq28_HTML.gif. To overcome this difficulty, we give some restrictions on upper and lower solutions such that their boundary values are zero. As far as the authors know, our three solutions theorem (Theorem 1.1 in Section 2) is new and first for singular p-Laplacian systems with weights of A B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq29_HTML.gif class.

To cover a larger class of differential system, we consider the systems of the form
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equ17_HTML.gif
where F , G : ( 0 , 1 ) × R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq30_HTML.gif are continuous. We give more conditions on F and G as follows: ( F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq31_HTML.gif) = For each t ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq32_HTML.gif, F ( t , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq33_HTML.gif and G ( t , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq34_HTML.gif are nondecreasing in u.; (H) = There exist h 1 , h 2 A B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq35_HTML.gif and f , g C ( R , R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq36_HTML.gif such that
0 lim s 0 f ( s ) φ p ( | s | ) < , 0 lim s 0 g ( s ) φ p ( | s | ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equc_HTML.gif
and
| F ( t , u ) | h 1 ( t ) f ( u ) , | G ( t , u ) | h 2 ( t ) g ( u ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equd_HTML.gif

for all t ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq37_HTML.gif and u R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq38_HTML.gif.;

( F 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq39_HTML.gif) = F ( t , u ) u > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq40_HTML.gif and G ( t , u ) u > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq41_HTML.gif, for all ( t , u ) ( 0 , 1 ) × R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq42_HTML.gif.

We now state our first main result related to three solutions theorem as follows. See for more details in Section 2.

Theorem 1.1 Assume (H), ( F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq31_HTML.gif) and ( F 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq43_HTML.gif). Let ( α 1 , α ¯ 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq44_HTML.gif, ( β 2 , β ¯ 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq45_HTML.gifbe a lower solution and an upper solution and ( α 2 , α ¯ 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq46_HTML.gif, ( β 1 , β ¯ 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq47_HTML.gifbe a strict lower solution and a strict upper solution of problem (P) respectively. Also, assume that all of them are contained in C 0 1 [ 0 , 1 ] × C 0 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq48_HTML.gifand satisfy ( α 1 , α ¯ 1 ) ( β 1 , β ¯ 1 ) ( β 2 , β ¯ 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq49_HTML.gif, ( α 1 , α ¯ 1 ) ( α 2 , α ¯ 2 ) ( β 2 , β ¯ 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq50_HTML.gif, ( α 2 , α ¯ 2 ) ( β 1 , β ¯ 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq51_HTML.gif. Then problem (P) has at least three solutions ( u 1 , v 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq52_HTML.gif, ( u 2 , v 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq53_HTML.gifand ( u 3 , v 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq54_HTML.gifsuch that ( α 1 , α ¯ 1 ) ( u 1 , v 1 ) ( β 1 , β ¯ 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq55_HTML.gif, ( α 2 , α ¯ 2 ) ( u 2 , v 2 ) ( β 2 , β ¯ 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq56_HTML.gif, ( α 1 , α ¯ 1 ) ( u 3 , v 3 ) ( β 2 , β ¯ 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq57_HTML.gifand ( u 3 , v 3 ) ( β 1 , β ¯ 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq58_HTML.gif, ( u 3 , v 3 ) ( α 2 , α ¯ 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq59_HTML.gif.

As an application of Theorem 1.1, we study the existence, nonexistence, and multiplicity of positive radial solutions for the following quasilinear system on an exterior domain:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equ18_HTML.gif

where Ω = { x R N : | x | > r 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq60_HTML.gif, r 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq61_HTML.gif, 1 < p < N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq62_HTML.gif, Δ p z = div ( | z | p 2 z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq63_HTML.gif, k i C ( [ r 0 , ) , ( 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq64_HTML.gif, i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq3_HTML.gif and f , g C ( R + , R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq65_HTML.gif with R + = [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq7_HTML.gif.

In recent years, the existence of positive solutions for such systems has been widely studied, for example, in [1] and [27] for second order ODE systems, in [3, 7, 9, 10, 13, 14, 16] and [8] for semilinear elliptic systems on a bounded domain and in [5, 15, 17] and [2] for p-Laplacian systems on a bounded domain.

For a precise description, let us give the list of assumptions that we consider.

(k) = k i KA KB https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq66_HTML.gif, where

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Eque_HTML.gif

( f 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq67_HTML.gif) = f 0 = lim s 0 + f ( s ) s p 1 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq68_HTML.gif and g 0 = lim s 0 + g ( s ) s p 1 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq69_HTML.gif,;

( f 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq70_HTML.gif) = lim s f ( ρ ( g ( s ) ) 1 p 1 ) s p 1 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq71_HTML.gif for all ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq72_HTML.gif,;

( f 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq73_HTML.gif) = f and g are nondecreasing..

Condition ( f 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq70_HTML.gif) is sometimes called a combined sublinear effect at ∞ and simple examples satisfying ( f 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq67_HTML.gif) ( f 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq73_HTML.gif) can be given as follows:
f ( w ) = { w r , w 1 , w q , w 1 , g ( z ) = { z γ , z 1 , z δ , z 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equf_HTML.gif
where r , γ > p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq74_HTML.gif and q δ < ( p 1 ) 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq75_HTML.gif, and also
{ f ( z ) = arctan ( z r ) , g ( w ) = w q , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equg_HTML.gif

where r , q > p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq76_HTML.gif.

Among the reference works mentioned above, Hai and Shivaji [17] and Ali and Shivaji [2] (with more general nonlinearities) considered problem ( P E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq77_HTML.gif) with case k i 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq25_HTML.gif and Ω bounded. For C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq26_HTML.gif monotone functions f and g with lim s f ( s ) = = lim s g ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq78_HTML.gif and satisfying condition ( f 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq70_HTML.gif), they proved that there exists λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq79_HTML.gif such that the problem has at least one positive solution for λ > λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq80_HTML.gif.

We first transform ( P E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq77_HTML.gif) into one-dimensional p-Laplacian systems ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif) with change of variables z ( r ) = z ( | x | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq81_HTML.gif, w ( r ) = w ( | x | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq82_HTML.gif, u ( t ) = z ( ( r r 0 ) N + p p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq83_HTML.gif and v ( t ) = w ( ( r r 0 ) N + p p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq84_HTML.gif where h i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq2_HTML.gif is given by
h i ( t ) = ( p 1 N p ) p r 0 p t p ( N 1 ) N p k i ( r 0 t ( p 1 ) N p ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equh_HTML.gif

It is not hard to see that if k i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq85_HTML.gif in ( P E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq77_HTML.gif) satisfies (k), then h i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq2_HTML.gif in ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif) satisfies h i A B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq86_HTML.gif, for i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq3_HTML.gif. Mainly by making use of Theorem 1.1, we prove the following existence result for problem ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif)

Theorem 1.2 Assume h i A B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq87_HTML.gif, i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq3_HTML.gif, ( f 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq67_HTML.gif), ( f 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq70_HTML.gif) and ( f 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq73_HTML.gif). Then there exists λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq88_HTML.gifsuch that ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif) has no positive solution for λ < λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq89_HTML.gif, at least one positive solution at λ = λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq90_HTML.gifand at least two positive solutions for λ > λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq91_HTML.gif.

As a corollary, we obtain our second main result as follows.

Corollary 1.3 Assume (k), ( f 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq67_HTML.gif), ( f 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq70_HTML.gif) and ( f 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq73_HTML.gif). Then there exists λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq88_HTML.gifsuch that ( P E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq77_HTML.gif) has no positive radial solution for λ < λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq89_HTML.gif, at least one positive radial solution at λ = λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq90_HTML.gifand at least two positive radial solutions for λ > λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq91_HTML.gif.

We finally notice that the first eigenfunctions of
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equ19_HTML.gif

make an important role to construct upper solutions in the proofs of Theorem 1.2 and Theorem 1.1. This is possible due to a recent work of Kajikiya, Lee, and Sim [19] which exploits the existence of discrete eigenvalues and the properties of corresponding eigenfunctions for problem (E) with h i A B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq87_HTML.gif.

This paper is organized as follows. In Section 2, we state a C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq26_HTML.gif-regularity result and a three solutions theorem for singular p-Laplacian systems. In addition, we introduce definitions of (strict) upper and lower solutions, a related theorem and a fixed point theorem for later use. In Section 3, we prove Theorem 1.2.

2 Three solutions theorem

In this section, we give definitions of upper and lower solutions and prove three solutions theorem for the following singular system
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equ20_HTML.gif

where F , G : ( 0 , 1 ) × R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq30_HTML.gif are continuous.

We call ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq92_HTML.gif a solution of (P) if ( u , v ) ( C [ 0 , 1 ] × C [ 0 , 1 ] ) ( C 1 ( 0 , 1 ) × C 1 ( 0 , 1 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq93_HTML.gif, ( φ p ( u ( t ) ) , φ p ( v ( t ) ) ) C 1 ( 0 , 1 ) × C 1 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq94_HTML.gif and ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq92_HTML.gif satisfies (P).

Definition 2.1 We say that ( α , α ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq95_HTML.gif is a lower solution of problem (P) if ( α , α ¯ ) ( C 1 ( 0 , 1 ) × C 1 ( 0 , 1 ) ) ( C [ 0 , 1 ] × C [ 0 , 1 ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq96_HTML.gif, ( φ p ( α ( t ) ) , φ p ( α ¯ ( t ) ) ) C 1 ( 0 , 1 ) × C 1 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq97_HTML.gif and
{ φ p ( α ( t ) ) + F ( t , α ¯ ( t ) ) 0 , t ( 0 , 1 ) , φ p ( α ¯ ( t ) ) + G ( t , α ( t ) ) 0 , t ( 0 , 1 ) , α ( 0 ) 0 , α ¯ ( 0 ) 0 , α ( 1 ) 0 , α ¯ ( 1 ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equi_HTML.gif

We also say that ( β , β ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq98_HTML.gif is an upper solution of problem (P) if ( β , β ¯ ) ( C 1 ( 0 , 1 ) × C 1 ( 0 , 1 ) ) ( C [ 0 , 1 ] × C [ 0 , 1 ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq99_HTML.gif, ( φ p ( β ( t ) ) , φ p ( β ¯ ( t ) ) ) C 1 ( 0 , 1 ) × C 1 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq100_HTML.gif and it satisfies the reverse of the above inequalities. We say that ( α , α ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq95_HTML.gif and ( β , β ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq98_HTML.gif are strict lower solution and strict upper solution of problem (P), respectively, if ( α , α ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq95_HTML.gif and ( β , β ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq98_HTML.gif are lower solution and upper solution of problem (P), respectively and satisfying φ p ( α ( t ) ) + F ( t , α ¯ ( t ) ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq101_HTML.gif, φ p ( α ¯ ( t ) ) + G ( t , α ( t ) ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq102_HTML.gif, φ p ( β ( t ) ) + F ( t , β ¯ ( t ) ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq103_HTML.gif, φ p ( β ¯ ( t ) ) + G ( t , β ( t ) ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq104_HTML.gif for t ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq37_HTML.gif.

We note that the inequality on R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq105_HTML.gif can be understood componentwise. Let D α β = { ( t , u , v ) | ( α ( t ) , α ¯ ( t ) ) ( u , v ) ( β ( t ) , β ¯ ( t ) ) , t ( 0 , 1 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq106_HTML.gif. Then the fundamental theorem on upper and lower solutions for problem (P) is given as follows. The proof can be done by obvious combination from Lee [20], Lee and Lee [21] and Lü and O’Regan [22].

Theorem 2.2 Let ( α , α ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq107_HTML.gifand ( β , β ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq108_HTML.gifbe a lower solution and an upper solution of problem (P) respectively such that

( a 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq109_HTML.gif) = ( α ( t ) , α ¯ ( t ) ) ( β ( t ) , β ¯ ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq110_HTML.gif, for all t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq111_HTML.gif.

Assume ( F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq31_HTML.gif). Also assume that there exist h F , h G A B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq112_HTML.gifsuch that( a 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq113_HTML.gif) = | F ( t , v ) | h F ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq114_HTML.gif, | G ( t , u ) | h G ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq115_HTML.gif, for all ( t , u , v ) D α β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq116_HTML.gif..Then problem (P) has at least one solution ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq92_HTML.gifsuch that
( α ( t ) , α ¯ ( t ) ) ( u ( t ) , v ( t ) ) ( β ( t ) , β ¯ ( t ) ) , for all t [ 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equj_HTML.gif

Remark 2.3 It is not hard to see that condition (H) implies the following condition;

For each M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq117_HTML.gif, there exists C M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq118_HTML.gif such that
| F ( t , u ) | C M h 1 ( t ) φ p ( | u | ) , | G ( t , u ) | C M h 2 ( t ) φ p ( | u | ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equk_HTML.gif

for t ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq32_HTML.gif and | u | M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq119_HTML.gif.

Lemma 2.4 Assume (H) and ( F 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq39_HTML.gif). Let ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq92_HTML.gifbe a nontrivial solution of (P). Then there exists a > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq120_HTML.gifsuch that both u and v have no interior zeros in ( 0 , a ] [ 1 a , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq121_HTML.gif.

Proof Let ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq92_HTML.gif be a nontrivial solution of (P). Suppose, on the contrary, that there exist sequences ( t n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq122_HTML.gif, ( s n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq123_HTML.gif of interior zeros of u and v respectively with t n , s n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq124_HTML.gif. We note that both sequences should exist simultaneously. Indeed, if one of the sequences say, ( t n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq125_HTML.gif, does not exist, then assuming without loss of generality, u > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq126_HTML.gif on ( 0 , a ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq127_HTML.gif for some a > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq120_HTML.gif, we get φ p ( v ( s ) ) = G ( t , u ( t ) ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq128_HTML.gif for t ( 0 , a ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq129_HTML.gif by ( F 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq39_HTML.gif). From the monotonicity of φ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq130_HTML.gif, we know that v is concave on the interval. Thus v should have at most one interior zero in ( 0 , a ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq127_HTML.gif, a contradiction. With this concave-convex argument, we know that ( t n , t n 1 ) ( s n , s n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq131_HTML.gif, u v 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq132_HTML.gif on ( t n , t n 1 ) ( s n , s n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq133_HTML.gif and if t n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq134_HTML.gif and s n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq135_HTML.gif are local extremal points of u and v on ( t n , t n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq136_HTML.gif and ( s n , s n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq137_HTML.gif respectively, thus both t n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq134_HTML.gif and s n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq135_HTML.gif are in ( t n , t n 1 ) ( s n , s n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq133_HTML.gif. We consider the case that t n s n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq138_HTML.gif, t n s n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq139_HTML.gif and u , v > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq140_HTML.gif in ( t n , t n 1 ) ( s n , s n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq133_HTML.gif. All other cases can be explained by the same argument. If M = max { u , v } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq141_HTML.gif, then by using Remark 2.3, we have
u ( t n ) = t n t n φ p 1 ( s t n F ( r , v ( r ) ) d r ) d s t n t n φ p 1 ( s n t n F ( r , v ( r ) ) d r ) d s C M t n t n φ p 1 ( s n t n h 1 ( r ) v ( r ) p 1 d r ) d s C M ( t n t n φ p 1 ( s n t n h 1 ( r ) d r ) d s ) v ( t n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equ1_HTML.gif
(2.1)
and similarly,
v ( t n ) C M ( s n t n φ p 1 ( s s n h 2 ( r ) d r ) d s ) u ( t n ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equ2_HTML.gif
(2.2)
Therefore, it follows from plugging (2.2) into (2.1) that
u ( t n ) ( C M ) 2 ( t n t n φ p 1 ( s n t n h 1 ( r ) d r ) d s ) ( s n t n φ p 1 ( s s n h 2 ( r ) d r ) d s ) u ( t n ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equ3_HTML.gif
(2.3)
Since h i A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq142_HTML.gif, for sufficiently large n, we obtain
( C M ) 2 ( t n t n φ p 1 ( s n t n h 1 ( r ) d r ) d s ) ( s n t n φ p 1 ( s s n h 2 ( r ) d r ) d s ) < 1 / 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equl_HTML.gif

This contradicts (2.3) and the proof is done. □

Theorem 2.5 Assume (H) and ( F 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq39_HTML.gif). If ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq92_HTML.gifis a solution of (P), then ( u , v ) C 0 1 [ 0 , 1 ] × C 0 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq143_HTML.gif.

Proof Let ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq92_HTML.gif be a nontrivial solution of (P). Then u , v C 0 [ 0 , 1 ] C 1 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq144_HTML.gif so that it is enough to show
| u ( 0 + ) | < , | u ( 1 ) | < , | v ( 0 + ) | < , | v ( 1 ) | < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equm_HTML.gif
We will show | u ( 0 + ) | < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq145_HTML.gif. Other facts can be proved by the same manner. Suppose | u ( 0 + ) | = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq146_HTML.gif. By Lemma 2.4 and the concave-convex argument, we may assume without loss of generality that there exists a ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq147_HTML.gif such that u , v , u , v > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq148_HTML.gif on ( 0 , a ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq127_HTML.gif. Then for given ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq149_HTML.gif, by the fact h i B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq150_HTML.gif, i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq3_HTML.gif, there exists δ ( 0 , a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq151_HTML.gif such that
0 δ t p 1 h i ( t ) d t < ε , i = 1 , 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equn_HTML.gif
Let M = max { u , v } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq141_HTML.gif. Then integrating (P) over ( s , t ) ( 0 , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq152_HTML.gif and using Remark 2.3, we have
u ( s ) p 1 u ( t ) p 1 + C M s t h 1 ( τ ) ( v ( τ ) τ ) p 1 τ p 1 d τ u ( t ) p 1 + C M ( v ( s ) s ) p 1 s t h 1 ( τ ) τ p 1 d τ u ( t ) p 1 + C M ε ( v ( s ) s ) p 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equ4_HTML.gif
(2.4)
where we use the fact that ( v ( τ ) τ ) p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq153_HTML.gif is decreasing since v is concave. From u ( 0 + ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq154_HTML.gif and (2.4), we know lim s 0 + ( v ( s ) s ) p 1 = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq155_HTML.gif. This implies that conditions u ( 0 + ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq154_HTML.gif and v ( 0 + ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq156_HTML.gif are equivalent. From (2.4), we have
( s u ( s ) v ( s ) ) p 1 ( s v ( s ) ) p 1 u ( t ) p 1 + C M ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equo_HTML.gif
Thus we have
lim sup s 0 + ( s u ( s ) v ( s ) ) p 1 C M ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equp_HTML.gif
Since ε is arbitrary, we have
lim sup s 0 + ( s u ( s ) v ( s ) ) p 1 = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equ5_HTML.gif
(2.5)
Using the fact v ( 0 + ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq156_HTML.gif, with same argument, we have
lim sup s 0 + ( s v ( s ) u ( s ) ) p 1 = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equ6_HTML.gif
(2.6)
On the other hand, we observe the inequality
( α + β ) 1 p 1 C p ( α 1 p 1 + β 1 p 1 ) , for α , β 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equ7_HTML.gif
(2.7)
where
C p = { 1 if p 2 , 2 2 p p 1 if 1 < p < 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equq_HTML.gif
Since h i A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq157_HTML.gif, we may choose b ( 0 , min { a , 1 2 } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq158_HTML.gif such that
( C M ) 1 p 1 C p 0 b ( s 1 2 h i ( τ ) d τ ) 1 p 1 d s < 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equ8_HTML.gif
(2.8)
Integrating (P) over ( s , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq159_HTML.gif with 0 < s < t < b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq160_HTML.gif and using Remark 2.3, we get
u ( s ) p 1 u ( t ) p 1 + C M v ( t ) p 1 s t h 1 ( τ ) d τ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equr_HTML.gif
here we use the fact that v ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq161_HTML.gif is increasing in ( 0 , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq162_HTML.gif. Using (2.7), we have
u ( s ) C p u ( t ) + ( C M ) 1 p 1 C p v ( t ) ( s 1 2 h 1 ( τ ) d τ ) 1 p 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equ9_HTML.gif
(2.9)
Integrating (2.9) over ( 0 , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq163_HTML.gif with respect to s and using (2.8), we have
u ( t ) C p t u ( t ) + ( C M ) 1 p 1 C p v ( t ) 0 t ( s 1 2 h 1 ( τ ) d τ ) 1 p 1 d s C p t u ( t ) + 1 2 v ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equ10_HTML.gif
(2.10)
Similarly, we have
v ( t ) C p t v ( t ) + 1 2 u ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equ11_HTML.gif
(2.11)
Adding (2.10) and (2.11), we have
0 < 1 2 C p < t u ( t ) + t v ( t ) u ( t ) + v ( t ) t u ( t ) v ( t ) + t v ( t ) u ( t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equ12_HTML.gif
(2.12)

on ( 0 , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq162_HTML.gif. From (2.5) and (2.6), we see that the right-hand side of (2.12) goes to zero as t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq164_HTML.gif. This is a contradiction and the proof is complete. □

Now, we consider the three solutions theorem for singular p-Laplacian system (P). For ν L 1 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq165_HTML.gif, if
ζ ( x ) = 0 1 φ p 1 ( x 0 s ν ( τ ) d τ ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equs_HTML.gif
then the zero of ζ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq166_HTML.gif, denoted by ξ ( ν ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq167_HTML.gif is uniquely determined by ν. Define A : L 1 ( 0 , 1 ) C 0 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq168_HTML.gif by taking
A ( ν ) ( t ) = 0 t φ p 1 ( ξ ( ν ) 0 s ν ( τ ) d τ ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equt_HTML.gif
It is known that A is completely continuous [24]. Define X C 0 1 [ 0 , 1 ] × C 0 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq169_HTML.gif with norm ( u , v ) X = u + v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq170_HTML.gif. We note that
| u ( t ) | 2 t ( 1 t ) u , for all u C 0 1 [ 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equ13_HTML.gif
(2.13)
If F and G satisfy condition (H), then for ( u , v ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq171_HTML.gif, from Remark 2.3 and (2.13), we get
0 1 | F ( t , v ( t ) ) | d t 0 1 h 1 ( t ) f ( v ( t ) ) d t 0 1 h 1 ( t ) C 0 | v ( t ) | p 1 d t 2 p 1 C 0 v p 1 0 1 t p 1 ( 1 t ) p 1 h 1 ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equu_HTML.gif
This implies F ( , v ( ) ) L 1 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq172_HTML.gif and by similar computation, we also get G ( , u ( ) ) L 1 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq173_HTML.gif. This fact enables us to define the integral operator for problem (P) and the regularity of solutions (Theorem 2.5) is crucial in this argument. Now, define an operator T by
T ( u , v ) = ( A ( F ( t , v ( t ) ) ) , A ( G ( t , u ( t ) ) ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equv_HTML.gif

then we see that T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq174_HTML.gif and completely continuous.

Lemma 2.6 Assume (H), ( F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq31_HTML.gif) and ( F 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq39_HTML.gif). Let ( α , α ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq95_HTML.gifand ( β , β ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq98_HTML.gifbe a strict lower solution and a strict upper solution of problem (P) respectively such that ( α , α ¯ ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq175_HTML.gif, ( β , β ¯ ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq176_HTML.gifand ( α , α ¯ ) ( β , β ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq177_HTML.gif. Then problem (P) has at least one solution ( u , v ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq178_HTML.gifsuch that
( α , α ¯ ) ( u , v ) ( β , β ¯ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equw_HTML.gif
Moreover, for R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq179_HTML.giflarge enough,
deg ( I T , Ω , 0 ) = 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equx_HTML.gif

where Ω = { ( u , v ) X | ( α , α ¯ ) ( u , v ) ( β , β ¯ ) , ( u , v ) X < R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq180_HTML.gif.

Proof Define γ : [ 0 , 1 ] × R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq181_HTML.gif given by
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equy_HTML.gif
and also define
F ( t , v ( t ) ) = F ( t , γ ¯ ( t , v ( t ) ) ) , G ( t , u ( t ) ) = G ( t , γ ( t , u ( t ) ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equz_HTML.gif
Let us consider the following modified problem
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equ21_HTML.gif
We first show that there exists a constant R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq179_HTML.gif such that if ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq92_HTML.gif is a solution of ( P ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq182_HTML.gif), then ( u , v ) Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq183_HTML.gif. In fact, every solution ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq92_HTML.gif of ( P ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq182_HTML.gif) satisfies ( α , α ¯ ) ( u , v ) ( β , β ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq184_HTML.gif on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq185_HTML.gif. From (H), ( F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq31_HTML.gif) and the fact that ( α , α ¯ ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq175_HTML.gif, ( β , β ¯ ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq186_HTML.gif, we get
| φ p ( u ( t ) ) | = | t 0 t F ( τ , v ( τ ) ) d τ | t 0 t max { | F ( τ , α ¯ ( τ ) ) | , | F ( τ , β ¯ ( τ ) ) | } d τ 0 1 h 1 ( t ) max t [ 0 , 1 ] { f ( α ¯ ( t ) ) , f ( β ¯ ( t ) ) } d t 0 1 c 1 h 1 ( t ) max t [ 0 , 1 ] { | α ¯ ( t ) | p 1 , | β ¯ ( t ) | p 1 } d t 2 p 1 c 1 max { α ¯ p 1 , β ¯ p 1 } 0 1 t p 1 ( 1 t ) p 1 h 1 ( t ) d t < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equaa_HTML.gif
Similarly, we see that v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq187_HTML.gif is bounded. Therefore, ( u , v ) X < R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq188_HTML.gif, for some R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq179_HTML.gif. Thus it is enough to show that
( α , α ¯ ) ( u , v ) ( β , β ¯ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equab_HTML.gif
Assume, on the contrary, that there exists t 0 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq189_HTML.gif such that
min ( u ( t ) α ( t ) ) = u ( t 0 ) α ( t 0 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equac_HTML.gif
Then choosing t 1 ( t 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq190_HTML.gif with ( u α ) ( t 1 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq191_HTML.gif, we get the following contradiction:
0 [ φ p ( u ( t 1 ) ) φ p ( α ( t 1 ) ) ] [ φ p ( u ( t 0 ) ) φ p ( α ( t 0 ) ) ] = t 0 t 1 F ( t , v ( t ) ) φ p ( α ( t ) ) d t t 0 t 1 F ( t , α ¯ ( t ) ) φ p ( α ( t ) ) d t < 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equad_HTML.gif
Now, assume u ( 0 ) = α ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq192_HTML.gif. Since u ( t ) > α ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq193_HTML.gif on t ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq32_HTML.gif and u ( 0 ) = α ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq194_HTML.gif, there exists t 2 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq195_HTML.gif such that u ( t 2 ) α ( t 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq196_HTML.gif and we get the same contradiction from the above calculation by using 0 instead of t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq197_HTML.gif. For u ( 1 ) = α ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq198_HTML.gif case, we also get the same contradiction. Consequently, we get α u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq199_HTML.gif. The other cases can be proved by the same manner. Taking Ω = { ( u , v ) X | ( α , α ¯ ) ( u , v ) ( β , β ¯ ) , ( u , v ) X < R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq200_HTML.gif, we see that every solution of ( P ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq182_HTML.gif) is contained in Ω. We now compute deg ( I T , Ω , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq201_HTML.gif. For this purpose, let us consider the operator T ¯ : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq202_HTML.gif defined by
T ¯ ( u , v ) ( t ) = ( A ( F ( t , v ( t ) ) ) , A ( G ( t , u ( t ) ) ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equae_HTML.gif
Then it is obvious that T ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq203_HTML.gif is completely continuous. We show that there exists R ¯ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq204_HTML.gif such that R ¯ > R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq205_HTML.gif and T ¯ ( X ) B ( 0 , R ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq206_HTML.gif. Indeed, since A ( F ( 0 , v ( 0 ) ) ) = 0 = A ( F ( 1 , v ( 1 ) ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq207_HTML.gif, there is t ˜ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq208_HTML.gif such that d d t A ( F ( t , v ( t ) ) ) | t = t ˜ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq209_HTML.gif. By integrating
d d t φ p ( d d t A ( F ( t , v ( t ) ) ) ) = F ( t , v ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equaf_HTML.gif
from t ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq210_HTML.gif to t, we have
| φ p ( d d t A ( F ( t , v ( t ) ) ) ) | = | t ˜ t F ( τ , v ( τ ) ) d τ | 0 1 h 1 ( t ) f ( γ ¯ ( t , v ( t ) ) ) d t 0 1 h 1 ( t ) C 1 | γ ¯ ( t , v ( t ) ) | p 1 d t 0 1 h 1 ( t ) C 1 max { | β ¯ ( t ) | p 1 , | α ¯ ( t ) | p 1 } d t C 2 max { β ¯ p 1 , α ¯ p 1 } 0 1 t p 1 ( 1 t ) p 1 h 1 ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equag_HTML.gif
Similarly, we see that d d t A ( G ( t , u ( t ) ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq211_HTML.gif is bounded. Therefore, we get
deg ( I T ¯ , B ( 0 , R ¯ ) , 0 ) = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equah_HTML.gif
Since every solution of ( P ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq182_HTML.gif) is contained in Ω, the excision property implies that
deg ( I T ¯ , Ω , 0 ) = deg ( I T ¯ , B ( 0 , R ¯ ) , 0 ) = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equai_HTML.gif
Since T ¯ = T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq212_HTML.gif on Ω, we finally get
deg ( I T , Ω , 0 ) = deg ( I T ¯ , Ω , 0 ) = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equaj_HTML.gif

This completes the proof. □

We now prove three solutions theorem for (P).

Proof of Theorem 1.1

Define
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equak_HTML.gif
and let us consider
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equ22_HTML.gif
Then noting that every solution ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq92_HTML.gif of ( P ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq213_HTML.gif) satisfies ( α 1 , α ¯ 1 ) ( u , v ) ( β 2 , β ¯ 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq214_HTML.gif, we may choose K 1 , K 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq215_HTML.gif, by (H) such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equal_HTML.gif
Let λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq216_HTML.gif and μ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq217_HTML.gif be the first eigenvalues of
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equ23_HTML.gif
for i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq3_HTML.gif respectively and let e 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq218_HTML.gif and e 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq219_HTML.gif be corresponding eigenfunctions with e 1 = e 2 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq220_HTML.gif. Since e 1 , e 2 C 0 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq221_HTML.gif are positive and concave [19], we may choose M 1 , M 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq222_HTML.gif such that ( M 1 e 1 , M 2 e 2 ) ( β 2 , β ¯ 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq223_HTML.gif ( M 1 e 1 , M 2 e 2 ) ( α 1 , α ¯ 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq224_HTML.gif and for t ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq32_HTML.gif
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equam_HTML.gif
We show that ( M 1 e 1 , M 2 e 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq225_HTML.gif and ( M 1 e 1 , M 2 e 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq226_HTML.gif are a strict upper solution and a strict lower solution of ( P ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq213_HTML.gif) respectively. Indeed,
φ p ( M 1 e 1 ( t ) ) + F ( t , γ ¯ 1 ( t , M 2 e 2 ( t ) ) ) = φ p ( M 1 e 1 ( t ) ) + F ( t , β ¯ 2 ( t ) ) λ 1 h 1 ( t ) φ p ( M 1 e 1 ( t ) ) + h 1 ( t ) f ( β ¯ 2 ( t ) ) λ 1 h 1 ( t ) φ p ( M 1 e 1 ( t ) ) + h 1 ( t ) K 1 φ p ( | β ¯ 2 ( t ) | ) < 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equan_HTML.gif
Similarly, we get
φ p ( M 2 e 2 ( t ) ) + G ( t , γ 1 ( t , M 1 e 1 ( t ) ) ) < 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equao_HTML.gif
Moreover,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equap_HTML.gif
Similarly, we also get
φ p ( M 2 e 2 ( t ) ) + F ( t , γ ¯ 1 ( t , M 2 e 2 ( t ) ) ) > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equaq_HTML.gif
For R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq179_HTML.gif, large enough, define
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equar_HTML.gif
Then by Theorem 2.2, there exist two solutions ( u 1 , v 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq227_HTML.gif and ( u 2 , v 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq228_HTML.gif of (P) satisfying ( α 1 , α ¯ 1 ) ( u 1 , v 1 ) ( β 1 , β ¯ 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq55_HTML.gif and ( α 2 , α ¯ 2 ) ( u 2 , v 2 ) ( β 2 , β ¯ 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq229_HTML.gif. Therefore, by Lemma 2.6, we get
deg ( I T ˜ , Ω 1 , 0 ) = deg ( I T ˜ , Ω 2 , 0 ) = deg ( I T ˜ , Ω 3 , 0 ) = 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equas_HTML.gif
and by the excision property, we have
deg ( I T ˜ , Ω 3 ( Ω 1 Ω 2 ) , 0 ) = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equat_HTML.gif

This completes the proof.

3 Application

In this section, we prove the existence, nonexistence, and multiplicity of positive solutions for ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif) by using three solutions theorem in Section 2. Let us define a cone
K = { u C [ 0 , 1 ] | u are concave and u ( 0 ) = 0 = u ( 1 ) } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equau_HTML.gif
and define A λ , B λ : K C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq230_HTML.gif by taking
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equav_HTML.gif
where σ v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq231_HTML.gif and σ u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq232_HTML.gif are unique zeros of
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equaw_HTML.gif
respectively. And define T λ : K × K C [ 0 , 1 ] × C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq233_HTML.gif by
T λ ( u , v ) = ( A λ ( v ) , B λ ( u ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equax_HTML.gif

Then it is known that T λ : K × K K × K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq234_HTML.gif is completely continuous [25] and ( u , v ) = T λ ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq235_HTML.gif in K × K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq236_HTML.gif is equivalent to the fact that ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq92_HTML.gif is a positive solution of ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif). We know from Theorem 2.5 that under assumptions h i A B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq237_HTML.gif i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq3_HTML.gif and ( f 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq67_HTML.gif), any solution ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq92_HTML.gif of problem ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif) is in C 0 1 [ 0 , 1 ] × C 0 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq238_HTML.gif.

Remark 3.1 If ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq92_HTML.gif is a solution of ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif), then u = A λ ( B λ ( u ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq239_HTML.gif and v = B λ ( A λ ( v ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq240_HTML.gif.

For later use, we introduce the following well-known result. See [12] for proof and details.

Proposition 3.2 Let X be a Banach space, K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq241_HTML.gifan order cone in X. Assume that Ω 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq242_HTML.gifand Ω 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq243_HTML.gifare bounded open subsets in X with 0 Ω 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq244_HTML.gifand Ω ¯ 1 Ω 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq245_HTML.gif. Let A : K ( Ω ¯ 2 Ω 1 ) K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq246_HTML.gifbe a completely continuous operator such that either
  1. (i)

    A u u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq247_HTML.gif, u K Ω 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq248_HTML.gif, and A u u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq249_HTML.gif, u K Ω 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq250_HTML.gif

     
or
  1. (ii)

    A u u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq249_HTML.gif, u K Ω 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq248_HTML.gif, and A u u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq247_HTML.gif, u K Ω 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq250_HTML.gif.

     

Then A has a fixed point in K ( Ω ¯ 2 Ω 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq251_HTML.gif.

Lemma 3.3 Assume h i A B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq87_HTML.gif, i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq3_HTML.gif, ( f 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq70_HTML.gif) and ( f 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq73_HTML.gif). Let R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq252_HTML.gifbe a compact subset of ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq253_HTML.gif. Then there exists a constant b R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq254_HTML.gifsuch that for all λ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq255_HTML.gifand all possible positive solutions ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq92_HTML.gifof ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif), one has ( u , v ) b R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq256_HTML.gif.

Proof If it is not true, then there exist { λ n } R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq257_HTML.gif and solutions { ( u n , v n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq258_HTML.gif of ( P λ n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq259_HTML.gif) such that ( u n , v n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq260_HTML.gif. We note that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equay_HTML.gif
where Λ = max { λ 1 p 1 | λ R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq261_HTML.gif and
Q i = 0 1 2 φ p 1 ( s 1 2 h i ( τ ) d τ ) d s + 1 2 1 φ p 1 ( 1 2 s h i ( τ ) d τ ) d s , i = 1 , 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equaz_HTML.gif
This implies both u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq262_HTML.gif and v n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq263_HTML.gif. Moreover, by the above estimation,
u n Λ Q 1 φ p 1 ( f ( Λ Q 2 φ p 1 ( g ( u n ) ) ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equba_HTML.gif
Thus we get
1 φ p ( Λ Q 1 ) f ( Λ Q 2 φ p 1 ( g ( u n ) ) ) φ p ( u n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equbb_HTML.gif

as u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq262_HTML.gif and this contradiction completes the proof. □

Lemma 3.4 Assume h i A B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq87_HTML.gif, i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq3_HTML.gif, ( f 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq70_HTML.gif) and ( f 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq73_HTML.gif). If ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif) has a lower solution ( α , α ¯ ) C 0 1 [ 0 , 1 ] × C 0 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq264_HTML.giffor some λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq265_HTML.gif, then ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif) has a solution ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq92_HTML.gifsuch that ( α , α ¯ ) ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq266_HTML.gif.

Proof It suffices to show the existence of an upper solution ( β , β ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq267_HTML.gif of ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif) satisfying ( α , α ¯ ) ( β , β ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq268_HTML.gif. Let ϕ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq269_HTML.gif and i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq3_HTML.gif be positive solutions of
{ φ p ( u ( t ) ) + λ h i ( t ) = 0 , t ( 0 , 1 ) , u ( 0 ) = 0 , u ( 1 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equbc_HTML.gif

(Case I) Both f and g are bounded.

Since ϕ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq269_HTML.gif ( i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq3_HTML.gif) are positive concave functions and ( α , α ¯ ) C 0 1 [ 0 , 1 ] × C 0 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq270_HTML.gif, we may choose M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq117_HTML.gif such that M > max { f 1 p 1 , g 1 p 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq271_HTML.gif and ( M ϕ 1 , M ϕ 2 ) ( α , α ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq272_HTML.gif. We now show that ( β , β ¯ ) = ( M ϕ 1 , M ϕ 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq273_HTML.gif is an upper solution of ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif). In fact,
φ p ( β ( t ) ) + λ h 1 ( t ) f ( β ¯ ( t ) ) = M p 1 φ p ( ϕ 1 ( t ) ) + λ h 1 ( t ) f ( M ϕ 2 ( t ) ) = λ h 1 ( t ) [ f ( M ϕ 2 ( t ) ) M p 1 ] λ h 1 ( t ) [ f M p 1 ] 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equbd_HTML.gif
Similarly,
φ p ( β ¯ ( t ) ) + λ h 2 ( t ) g ( β ( t ) ) λ h 2 ( t ) [ g M p 1 ] 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Eqube_HTML.gif

(Case II) g ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq274_HTML.gif as u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq275_HTML.gif.

Using ( f 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq70_HTML.gif), choose M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq117_HTML.gif such that ( g ( M ϕ 1 ) ) 1 p 1 ϕ 2 α ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq276_HTML.gif, M ϕ 1 α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq277_HTML.gif and
f ( ϕ 2 ( g ( M ϕ 1 ) ) 1 p 1 ) ( M ϕ 1 ) p 1 1 ϕ 1 p 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equbf_HTML.gif
Let ( β , β ¯ ) = ( M ϕ 1 , ( g ( M ϕ 1 ) ) 1 p 1 ϕ 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq278_HTML.gif. Then
φ p ( β ( t ) ) + λ h 1 ( t ) f ( β ¯ ( t ) ) = M p 1 φ p ( ϕ 1 ( t ) ) + λ h 1 ( t ) f ( ( g ( M ϕ 1 ) ) 1 p 1 ϕ 2 ( t ) ) = λ h 1 ( t ) [ f ( ( g ( M ϕ 1 ) ) 1 p 1 ϕ 2 ( t ) ) M p 1 ] λ h 1 ( t ) [ f ( ϕ 2 ( g ( M ϕ 1 ) ) 1 p 1 ) M p 1 ] 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equbg_HTML.gif
And
φ p ( β ¯ ( t ) ) + λ h 2 ( t ) g ( β ( t ) ) = g ( M ϕ 1 ) φ p ( ϕ 2 ( t ) ) + λ h 2 ( t ) g ( M ϕ 1 ( t ) ) = λ h 2 ( t ) [ g ( M ϕ 1 ( t ) ) g ( M ϕ 1 ) ] λ h 2 ( t ) [ g ( M ϕ 1 ) g ( M ϕ 1 ) ] = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equbh_HTML.gif

Thus ( β , β ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq108_HTML.gif is an upper solution of ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif).

(Case III) g is bounded and f ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq279_HTML.gif as u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq275_HTML.gif.

Choose M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq117_HTML.gif such that ( f ( M ϕ 2 ) ) 1 p 1 ϕ 1 α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq280_HTML.gif, M > g 1 p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq281_HTML.gif and M ϕ 2 α ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq282_HTML.gif and let
( β , β ¯ ) = ( ( f ( M ϕ 2 ) ) 1 p 1 ϕ 1 , M ϕ 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equbi_HTML.gif
Then
φ p ( β ( t ) ) + λ h 1 ( t ) f ( β ¯ ( t ) ) = f ( M ϕ 2 ) φ p ( ϕ 1 ( t ) ) + λ h 1 ( t ) f ( M ϕ 2 ( t ) ) = λ h 1 ( t ) [ f ( M ϕ 2 ( t ) ) f ( M ϕ 2 ) ] λ h 1 ( t ) [ f ( M ϕ 2 ) f ( M ϕ 2 ) ] = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equbj_HTML.gif
And
φ p ( β ¯ ( t ) ) + λ h 2 ( t ) g ( β ( t ) ) = M p 1 φ p ( ϕ 2 ( t ) ) + λ h 2 ( t ) g ( ( f ( M ϕ 2 ) ) 1 p 1 ϕ 1 ( t ) ) = λ h 2 ( t ) [ g ( ( f ( M ϕ 2 ) ) 1 p 1 ϕ 1 ( t ) ) M p 1 ] λ h 2 ( t ) [ g M p 1 ] 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equbk_HTML.gif
Consequently, by Theorem 2.2, ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif) has a solution satisfying
( α , α ¯ ) ( u , v ) ( β , β ¯ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equbl_HTML.gif

 □

Lemma 3.5 Assume h i A B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq87_HTML.gif, i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq3_HTML.gif, ( f 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq67_HTML.gif), ( f 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq70_HTML.gif) and ( f 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq73_HTML.gif). Then there exists λ ¯ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq283_HTML.gifsuch that if ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif) has a positive solution ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq92_HTML.gif, then λ λ ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq284_HTML.gif.

Proof Let ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq92_HTML.gif be a positive solution of ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif). Without loss of generality, we may assume λ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq285_HTML.gif. From ( f 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq67_HTML.gif), we know that
lim x 0 + f ( ρ φ p 1 ( g ( u ) ) ) φ p ( u ) = 0 , for all ρ > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equ14_HTML.gif
(3.1)
From (3.1) and ( f 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq70_HTML.gif), we can choose M f > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq286_HTML.gif such that
f ( Q 2 φ p 1 ( g ( u ) ) ) M f φ p ( u ) , for all u > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equ15_HTML.gif
(3.2)
where Q i = 0 1 2 φ p 1 ( s 1 2 h i ( τ ) d τ ) d s + 1 2 1 φ p 1 ( 1 2 s h i ( τ ) d τ ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq287_HTML.gif, i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq3_HTML.gif. Using (3.2) and ( f 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq73_HTML.gif), we have
u Q 1 φ p 1 ( λ f ( B λ ( u ) ) ) Q 1 φ p 1 ( λ f ( Q 2 φ p 1 ( λ g ( u ) ) ) ) Q 1 φ p 1 ( λ f ( Q 2 φ p 1 ( g ( u ) ) ) ) Q 1 φ p 1 ( λ M f φ p ( u ) ) Q 1 φ p 1 ( λ M f ) u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equbm_HTML.gif
Thus we have
λ ¯ 1 φ p ( Q 1 ) M f λ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equbn_HTML.gif

 □

Lemma 3.6 Assume h i A B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq87_HTML.gif, i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq3_HTML.gif, ( f 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq67_HTML.gif), ( f 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq70_HTML.gif) and ( f 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq73_HTML.gif). Then for each R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq179_HTML.gif, there exists λ R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq288_HTML.gifsuch that for λ > λ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq289_HTML.gif, ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif) has a positive solution ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq92_HTML.gifwith u > R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq290_HTML.gifand v > R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq291_HTML.gif.

Proof We know that if ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq92_HTML.gif satisfies u = A λ ( B λ ( u ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq239_HTML.gif and v = B λ ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq292_HTML.gif, then ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq92_HTML.gif is a solution of ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif). Since A λ , B λ : K K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq293_HTML.gif are completely continuous, A λ B λ : K K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq294_HTML.gif is also completely continuous. Given R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq179_HTML.gif, choose
λ R = max { φ p ( 2 R Γ 2 ) 1 g ( R 4 ) , φ p ( 2 R Γ 1 ) 1 f ( R 4 ) } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equbo_HTML.gif
where Γ i = min t [ 1 4 , 3 4 ] { 1 4 t φ p 1 ( s t h i ( τ ) d τ ) d s + t 3 4 φ p 1 ( t s h i ( τ ) d τ ) d s } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq295_HTML.gif. Let Ω 1 = { u C [ 0 , 1 ] | u < R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq296_HTML.gif. If u Ω 1 K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq297_HTML.gif, then for t [ 1 4 , 3 4 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq298_HTML.gif, u ( t ) 1 4 u 1 4 R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq299_HTML.gif. From the definition of B λ R ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq300_HTML.gif, we know that B λ R ( u ) ( σ u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq301_HTML.gif is the maximum value of B λ R ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq300_HTML.gif on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq185_HTML.gif. If σ u [ 1 4 , 3 4 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq302_HTML.gif, then from the choice of λ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq303_HTML.gif, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equbp_HTML.gif
If σ u > 3 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq304_HTML.gif, then we have
B λ R ( u ) 1 4 3 4 φ p 1 ( s 3 4 λ R h 2 ( τ ) g ( R 4 ) d τ ) d s 1 2 Γ 2 φ p 1 ( λ R g ( R 4 ) ) R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equbq_HTML.gif
If σ u < 1 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq305_HTML.gif, then
B λ R ( u ) 1 4 3 4 φ p 1 ( 1 4 s λ R h 2 ( τ ) g ( R 4 ) d τ ) d s 1 2 Γ 2 φ p 1 ( λ R g ( R 4 ) ) R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equbr_HTML.gif
By the concavity of B λ R ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq300_HTML.gif, we get for t [ 1 4 , 3 4 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq306_HTML.gif,
B λ R ( u ) ( t ) 1 4 B λ R ( u ) 1 4 R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equ16_HTML.gif
(3.3)
By similar argument as the above, with (3.3), we may show that
A λ R ( B λ R ( u ) ) 1 2 Γ 1 φ p 1 ( λ R f ( R 4 ) ) R = u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equbs_HTML.gif
Let Q i = 0 1 2 φ p 1 ( s 1 2 h i ( τ ) d τ ) d s + 1 2 1 φ p 1 ( 1 2 s h i ( τ ) d τ ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq307_HTML.gif, i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq3_HTML.gif. For ε < 1 φ p ( Q 1 ) λ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq308_HTML.gif, from ( f 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq70_HTML.gif), we may choose R ˜ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq309_HTML.gif such that R ˜ > R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq310_HTML.gif and
f ( Q 2 φ p 1 ( λ R ) φ p 1 ( g ( R ˜ ) ) ) ε φ p ( R ˜ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equbt_HTML.gif
Let Ω 2 = { u C [ 0 , 1 ] | u < R ˜ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq311_HTML.gif, then Ω 1 ¯ Ω 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq312_HTML.gif and for u Ω 2 K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq313_HTML.gif,
A λ R ( B λ R ( u ) ) Q 1 φ p 1 ( λ R f ( Q 2 φ p 1 ( λ R ) φ p 1 ( g ( R ˜ ) ) ) ) Q 1 φ p 1 ( λ R ε φ p ( R ˜ ) ) Q 1 φ p 1 ( λ R ε ) R ˜ R ˜ = u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equbu_HTML.gif

By Proposition 3.2, ( P λ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq314_HTML.gif) has a positive solution ( u R , v R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq315_HTML.gif such that u R > R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq316_HTML.gif and v R > R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq317_HTML.gif. We know that ( u R , v R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq315_HTML.gif is a lower solution of ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif) for λ > λ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq318_HTML.gif and by Lemma 3.4, the proof is complete. □

We now prove one of the main results for this paper.

Proof of Theorem 1.2

From Lemma 3.6 and Lemma 3.5, we know that the set S = { λ > 0 | ( P λ ) has a positive solution } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq319_HTML.gif is not empty and λ = inf S > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq320_HTML.gif. By Lemma 3.3 and complete continuity of T, there exist sequences { λ n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq321_HTML.gif and { ( u n , v n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq322_HTML.gif such that λ n λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq323_HTML.gif and ( u n , v n ) ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq324_HTML.gif in K × K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq325_HTML.gif with ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq326_HTML.gif a solution of ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq327_HTML.gif). We claim that ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq326_HTML.gif is a nontrivial solution of ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq327_HTML.gif). Suppose that it is not true, then there exists a sequence of solutions ( u n , v n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq328_HTML.gif for ( P λ n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq259_HTML.gif) such that ( u n , v n ) ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq329_HTML.gif and λ n λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq330_HTML.gif. As in the proof of Lemma 3.3, we get
1 φ p ( Λ Q 1 ) f ( Λ Q 2 φ p 1 ( g ( u n ) ) ) φ p ( u n ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equbv_HTML.gif
But from ( f 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq67_HTML.gif), we have a contradiction to the fact that the right side of the above inequality converges to zero as u n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq331_HTML.gif. Thus ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq326_HTML.gif is a nontrivial solution of ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq332_HTML.gif). According to Lemma 3.4 and the definition of λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq333_HTML.gif, we know that ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif) has at least one positive solution at λ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq334_HTML.gif and no positive solution for λ < λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq89_HTML.gif. To prove the existence of the second positive solution of ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif) for λ > λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq335_HTML.gif, we will use Theorem 1.1. Let λ > λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq335_HTML.gif. Then we have ( α 1 , α ¯ 1 ) = ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq336_HTML.gif a lower solution of ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif) and ( α 2 , α ¯ 2 ) = ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq337_HTML.gif a strict lower solution of ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif) in C 0 1 [ 0 , 1 ] × C 0 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq48_HTML.gif satisfying ( α 2 , α ¯ 2 ) ( α 1 , α ¯ 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq338_HTML.gif. For upper solutions, let λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq216_HTML.gif and μ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq217_HTML.gif be the first eigenvalues of for i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq3_HTML.gif respectively and let e 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq218_HTML.gif and e 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq219_HTML.gif be corresponding eigenfunctions with e 1 = e 2 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq339_HTML.gif. Since e 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq218_HTML.gif and e 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq219_HTML.gif are in C 0 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq28_HTML.gif and positive [19], we may choose c 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq340_HTML.gif and c 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq341_HTML.gif such that
λ c 1 e 2 p 1 < λ 1 e 1 p 1 and λ c 2 e 1 p 1 < μ 1 e 2 p 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equbw_HTML.gif
Also by the fact f 0 = g 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq342_HTML.gif, there exists a > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq120_HTML.gif such that
f ( u ) c 1 u p 1 , g ( u ) c 2 u p 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equbx_HTML.gif
for all | u | a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq343_HTML.gif and
a e 1 ( t ) < α 2 ( t ) , a e 2 ( t ) < α ¯ 2 ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equby_HTML.gif
Let ( β 1 , β ¯ 1 ) = ( a e 1 , a e 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq344_HTML.gif. Then ( β 1 , β ¯ 1 ) ( α 2 , α ¯ 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq345_HTML.gif and it is a strict upper solution of ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif) in C 0 1 [ 0 , 1 ] × C 0 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq48_HTML.gif. Indeed,
φ p ( β 1 ( t ) ) + λ h 1 ( t ) f ( β ¯ 1 ( t ) ) = a p 1 φ p ( e 1 ( t ) ) + λ h 1 ( t ) f ( a e 2 ( t ) ) λ 1 h 1 ( t ) a p 1 φ p ( e 1 ( t ) ) + λ h 1 ( t ) c 1 a p 1 φ p ( e 2 ( t ) ) = a p 1 h 1 ( t ) [ λ c 1 φ p ( e 2 ( t ) ) λ 1 φ p ( e 1 ( t ) ) ] < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equbz_HTML.gif
and
φ p ( β ¯ 1 ( t ) ) + λ h 2 ( t ) g ( β 1 ( t ) ) = a p 1 φ p ( e 2 ( t ) ) + λ h 2 ( t ) g ( a e 1 ( t ) ) μ 1 h 2 ( t ) a p 1 φ p ( e 2 ( t ) ) + λ h 2 ( t ) c 2 a p 1 φ p ( e 1 ( t ) ) = a p 1 h 2 ( t ) [ λ c 2 φ p ( e 1 ( t ) ) μ 1 φ p ( e 2 ( t ) ) ] < 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equca_HTML.gif
Finally, from Lemma 3.6, there exists λ ¯ > λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq346_HTML.gif such that ( P λ ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq347_HTML.gif) has a positive solution ( u ¯ , v ¯ ) C 0 1 [ 0 , 1 ] × C 0 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq348_HTML.gif satisfying u ¯ > max { α 2 , β 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq349_HTML.gif and v ¯ > max { α ¯ 2 , β ¯ 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq350_HTML.gif. By using the concavity of solutions, it is easily verified that
( β 1 , β ¯ 1 ) ( u ¯ , v ¯ ) and ( α 2 , α ¯ 2 ) ( u ¯ , v ¯ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_Equcb_HTML.gif

Therefore, ( β 2 , β ¯ 2 ) = ( u ¯ , v ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq351_HTML.gif is an upper solution of ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif) in C 0 1 [ 0 , 1 ] × C 0 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq352_HTML.gif. Now by Theorem 1.1, ( P λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq10_HTML.gif) has at least two positive solutions ( u 1 , v 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq52_HTML.gif and ( u 2 , v 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq53_HTML.gif such that ( α 2 , α ¯ 2 ) ( u 1 , v 1 ) ( β 2 , β ¯ 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq353_HTML.gif and ( α 1 , α ¯ 1 ) ( u 2 , v 2 ) ( β 2 , β ¯ 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq354_HTML.gif and ( u 2 , v 2 ) ( β 1 , β ¯ 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq355_HTML.gif ( u 2 , v 2 ) ( α 2 , α ¯ 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq356_HTML.gif.

Declarations

Acknowledgements

The authors express their thanks to Professors Ryuji Kajikiya, Yuki Naito and Inbo Sim for valuable discussions related to C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-63/MediaObjects/13661_2012_Article_186_IEq26_HTML.gif-regularity of solutions and also thank to the referees for their careful reading and valuable remarks and suggestions. The first author was supported by Pusan National University Research Grant, 2011. The second author was supported by Mid-career Researcher Program (No. 2010-0000377) and Basic Science Research Program (No. 2012005767) through NRF grant funded by the MEST.

Authors’ Affiliations

(1)
Department of Mathematics Education, Pusan National University
(2)
Department of Mathematics, Pusan National University

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© Lee and Lee; licensee Springer 2012

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