Open Access

On positive solutions to equations involving the one-dimensional p-Laplacian

Boundary Value Problems20132013:125

DOI: 10.1186/1687-2770-2013-125

Received: 15 October 2012

Accepted: 29 April 2013

Published: 15 May 2013

Abstract

We consider equations involving the one-dimensional p-Laplacian

( | u ( t ) | p 2 u ( t ) ) + λ f ( u ( t ) ) = 0 , t ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equa_HTML.gif

with the Dirichlet boundary conditions. By using time map methods, we show how changes of the sign of f ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq1_HTML.gif lead to multiple positive solutions of the problem for sufficiently large λ.

MSC:34B10, 34B18.

Keywords

positive solutions one-dimensional p-Laplacian uniqueness time map

1 Introduction

Let f : [ 0 , ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq2_HTML.gif be continuous and change its sign. Let Ω be an open subset of R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq3_HTML.gif with smooth boundary Ω. The semi-positone problems and their special cases
Δ u + λ f ( u ) = 0 in  Ω , u = 0 on  Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ1_HTML.gif
(1.1)
and
u ( t ) + λ f ( u ( t ) ) = 0 , t ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ2_HTML.gif
(1.2)

(and their finite difference analogues) have been extensively studied since early 1980s. Several different approaches such as variational methods, bifurcation theory, lower and upper solutions method and quadrature arguments have been successfully applied to show the existence of multiple solutions. See Brown and Budin [1], Peitgen et al. [2], Peitgen and Schmitt [3], Hess [4], Ambrosetti and Hess [5], Cosner and Schmitt [6], Dancer and Schmitt [7], Espinoza [8], Anuradha and Shivaji [9], Anuradha et al. [10], de Figueiredo [11], Lin and Pai [12], Clément and Sweers [13] and the references therein.

Very recently, Loc and Schmitt [14] considered the problem
Δ p u + λ f ( u ) = 0 in  Ω , u = 0 on  Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ3_HTML.gif
(1.3)
where Δ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq4_HTML.gif is the p-Laplace operator for p ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq5_HTML.gif. They assumed that the nonlinearity f is a continuous function on , f ( 0 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq6_HTML.gif, and there exist 0 < a 1 < b 1 < a 2 < b 2 < < b m 1 < a m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq7_HTML.gif such that f 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq8_HTML.gif on ( a k , b k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq9_HTML.gif and f 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq10_HTML.gif on ( b k , a k + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq11_HTML.gif for every k = 1 , , m 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq12_HTML.gif. They proved that, for λ sufficiently large, if
a k a k + 1 f ( s ) d s > 0 for all  k { 1 , , m 1 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ4_HTML.gif
(1.4)
then the problem (1.3) has at least m 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq13_HTML.gif positive bounded solutions u 1 , , u m 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq14_HTML.gif which belong to the Sobolev space W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq15_HTML.gif and are such that u ( a k , a k + 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq16_HTML.gif for each k { 1 , , m 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq17_HTML.gif, where
u = max { | u ( x ) | x Ω ¯ } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equb_HTML.gif

In the special case that p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq18_HTML.gif and N = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq19_HTML.gif, Brown and Budin [1] applied the quadrature arguments to get the following more detailed results.

Theorem A [[1], Theorem 3.8]

Assume that

(H1) f C 1 [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq20_HTML.gif;

(H2) f ( 0 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq21_HTML.gif;

(H3) There exists a 1 , , a n R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq22_HTML.gif such that 0 < a 1 < a 2 < < a n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq23_HTML.gif and f ( a i ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq24_HTML.gif for i = 1 , 2 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq25_HTML.gif;

(H4) If F ( u ) = 0 u f ( s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq26_HTML.gif, there exist b 1 , , b n 1 R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq27_HTML.gif with a 1 < b 1 < a 2 < b 2 < < a n 1 < b n 1 < a n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq28_HTML.gif such that f ( b i ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq29_HTML.gif and F ( b i ) > F ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq30_HTML.gif for 0 u b i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq31_HTML.gif, i = 1 , 2 , , n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq32_HTML.gif.

Then:
  1. (a)

    For all λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq33_HTML.gif, there exists a solution ( λ , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq34_HTML.gif of (1.2).

     
  2. (b)
    If λ > inf { λ ( ρ ) : ρ ( α i , β i ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq35_HTML.gif, there exist at least two solutions ( λ , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq34_HTML.gif of (1.2) such that
    α i < u < β i , i = 1 , 2 , , n 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equc_HTML.gif
     
where
β i = inf { u > b i f ( u ) = 0 } , α i = inf { u ( u , β i ) S } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ5_HTML.gif
(1.5)
and
S = { u u > 0 , f ( u ) > 0 , F ( u ) > F ( s ) for all s : 0 s < u } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ6_HTML.gif
(1.6)
  1. (c)
    If ( λ , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq34_HTML.gif is any solution of (1.2) such that α i < u < β i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq36_HTML.gif, then
    λ > 4 α i k 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equd_HTML.gif
     

where k = sup { | f ( u ) | : 0 u β i } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq37_HTML.gif.

Of course the natural question is whether or not the similar results still hold for the corresponding problem involving the one-dimensional p-Laplacian
( | u ( t ) | p 2 u ( t ) ) + λ f ( u ( t ) ) = 0 , t ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ7_HTML.gif
(1.7)

We shall answer these questions in the affirmative if p ( 1 , 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq38_HTML.gif. More precisely, we get the following theorem.

Theorem 1.1 Let p ( 1 , 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq38_HTML.gif and let (H1), (H3), (H4) hold. Assume that

(H2′) either f ( 0 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq21_HTML.gif or f ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq39_HTML.gif and
f 0 = lim s 0 + f ( s ) s p 1 > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ8_HTML.gif
(1.8)
Then:
  1. (a)
    For all λ > λ 1 f 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq40_HTML.gif, there exists a solution ( λ , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq34_HTML.gif of (1.7), and λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq41_HTML.gif is the least eigenvalue of BVP
    ( | u ( t ) | p 2 u ( t ) ) + λ | u ( t ) | p 2 u ( t ) = 0 , t ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ9_HTML.gif
    (1.9)
     
  2. (b)
    If λ > inf { λ ( ρ ) : ρ ( α i , β i ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq35_HTML.gif, there exist at least two solutions ( λ , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq34_HTML.gif of (1.7) such that
    α i < u < β i , i = 1 , 2 , , n 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Eque_HTML.gif
     
  3. (c)
    If ( λ , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq34_HTML.gif is any solution of (1.7) such that α i < u < β i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq36_HTML.gif, then
    λ > ( α i C ) p 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equf_HTML.gif
     
where
C = p 1 p ( 1 2 ) p p 1 ( sup s [ 0 , β i ] | f ( s ) | ) 1 p 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ10_HTML.gif
(1.10)

We shall apply the time map method to show how changes of the sign of f ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq1_HTML.gif lead to multiple positive solutions of (1.7) for sufficiently large λ.

In the following, we extend f so that f ( u ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq42_HTML.gif for all u < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq43_HTML.gif, then all the solutions of (1.7) are positive on ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq44_HTML.gif.

2 Preliminaries

To prove our main results, we use the uniqueness results due to Reichel and Walter [15] on the initial value problem
( | u ( t ) | p 2 u ( t ) ) + λ f ( u ( t ) ) = 0 , t ( 0 , 1 ) , u ( a ) = b , u ( a ) = d , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ11_HTML.gif
(2.1)

where a [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq45_HTML.gif and b , d R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq46_HTML.gif.

Lemma 2.1 Let (H1) hold. If a ( 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq47_HTML.gif and d 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq48_HTML.gif, then the initial value problem (2.1) has a unique local solution. The extension u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq49_HTML.gif remains unique as long as u ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq50_HTML.gif.

Proof It is an immediate consequence of Reichel and Walter [[15], Theorem 2]. □

Lemma 2.2 Let (H1) hold. Let a ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq51_HTML.gif, and let ρ ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq52_HTML.gif be such that
f ( ρ ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equg_HTML.gif
Then the initial value problem
( | u ( t ) | p 2 u ( t ) ) + λ f ( u ( t ) ) = 0 , t ( 0 , 1 ) , u ( a ) = ρ , u ( a ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ12_HTML.gif
(2.2)

has a unique local solution.

Proof (H1) implies that f is locally Lipschitzian. This together with the assumption f ( ρ ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq53_HTML.gif and using [[15], (iii) and (v) in the case (β) of Theorem 4] yields that (2.2) has a unique solution in some neighborhood of a. □

Lemma 2.3 Let g : R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq54_HTML.gif be continuous. Let u be a solution of the equation
( | u ( t ) | p 2 u ( t ) ) + g ( u ( t ) ) = 0 , t ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ13_HTML.gif
(2.3)
with u = ρ S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq55_HTML.gif. Let x 0 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq56_HTML.gif be such that u ( x 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq57_HTML.gif. Then
u ( x 0 t ) u ( x 0 + t ) , t ( 0 , min { x 0 , 1 x 0 } ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ14_HTML.gif
(2.4)
Proof Since g is independent of t, both u ( x 0 t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq58_HTML.gif and u ( x 0 + t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq59_HTML.gif satisfy the initial value problem
{ ( | w ( t ) | p 2 w ( t ) ) + g ( w ( t ) ) = 0 , t ( 0 , min { x 0 , 1 x 0 } ) , w ( 0 ) = u ( x 0 ) , w ( 0 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ15_HTML.gif
(2.5)

By Lemmas 2.1 and 2.2, (2.5) has a unique solution defined on t ( 0 , min { x 0 , 1 x 0 } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq60_HTML.gif. Therefore, (2.4) is true. □

Lemma 2.4 Let ( λ , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq61_HTML.gif be a positive solution of the problem
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ16_HTML.gif
(2.6)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ17_HTML.gif
(2.7)
with u = ρ S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq55_HTML.gif and λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq33_HTML.gif. Let x 0 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq62_HTML.gif be such that u ( x 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq57_HTML.gif. Then
  1. (a)

    x 0 = 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq63_HTML.gif;

     
  2. (b)

    x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq64_HTML.gif is the unique point on which u attains its maximum;

     
  3. (c)

    u ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq65_HTML.gif, t ( 0 , 1 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq66_HTML.gif.

     
Proof (a) Suppose on the contrary that x 0 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq67_HTML.gif, say x 0 > 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq68_HTML.gif, then
0 = u ( 1 ) = u ( 1 2 x 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equh_HTML.gif
However, this is impossible since 1 2 x 0 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq69_HTML.gif and u > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq70_HTML.gif in ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq44_HTML.gif. Therefore x 0 = 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq63_HTML.gif.
  1. (b)
    Suppose on the contrary that there exists x 1 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq71_HTML.gif with x 1 x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq72_HTML.gif and
    u ( x 1 ) = u ( x 0 ) = : ρ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equi_HTML.gif
     

We may assume that x 1 < x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq73_HTML.gif. The other case can be treated in a similar way.

If u ( t ) u ( x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq74_HTML.gif in the interval ( x 1 , x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq75_HTML.gif, then Lemma 2.3 yields that
u ( t ) u ( x 0 ) = ρ > 0 , t ( 0 , 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equj_HTML.gif

This contradicts the boundary conditions u ( 0 ) = u ( 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq76_HTML.gif. Therefore, u ( t ) u ( x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq77_HTML.gif in any subinterval of ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq44_HTML.gif.

So, there exists x ( x 1 , x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq78_HTML.gif, such that
u ( x ) = min { u ( t ) t ( x 1 , x 0 ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equk_HTML.gif
Obviously,
0 < u ( x ) < ρ , u ( x ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equl_HTML.gif
Multiplying both sides of the equation in (2.6) by u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq79_HTML.gif and integrating from t to x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq64_HTML.gif, we get that
| u ( t ) | p = λ p p 1 [ F ( ρ ) F ( u ( t ) ) ] , t [ 0 , 1 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ18_HTML.gif
(2.8)
and subsequently,
0 = | u ( x ) | p = λ p p 1 [ F ( ρ ) F ( u ( x ) ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equm_HTML.gif
This contradicts the facts that ρ S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq80_HTML.gif and u ( x ) < ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq81_HTML.gif. Therefore,
u ( 1 2 ) > u ( t ) , t [ 0 , 1 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equn_HTML.gif
Similarly, we can prove that
u ( 1 2 ) > u ( t ) , t ( 1 2 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equo_HTML.gif
  1. (c)
    Suppose on the contrary that there exists x ˆ ( 0 , 1 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq82_HTML.gif with u ( x ˆ ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq83_HTML.gif. Then
    u ( x ˆ ) < ρ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equp_HTML.gif
     
This together with (2.8) implies that
0 = | u ( x ˆ ) | p = λ p p 1 [ F ( ρ ) F ( u ( x ˆ ) ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equq_HTML.gif

This contradicts the facts that ρ S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq80_HTML.gif and u ( x ˆ ) < ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq84_HTML.gif. □

3 Proof of the main results

To prove Theorem 1.1, we need the following preliminary results.

Lemma 3.1 For any ρ S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq80_HTML.gif, there exists a unique λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq33_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ19_HTML.gif
(3.1)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ20_HTML.gif
(3.2)

has a positive solution ( λ , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq34_HTML.gif with u = ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq85_HTML.gif. Moreover, ρ λ ( ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq86_HTML.gif is a continuous function on S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq87_HTML.gif.

Proof By Lemma 2.4, ( λ , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq34_HTML.gif is a positive solution of (3.1), (3.2) if and only if ( λ , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq34_HTML.gif is a positive solution of
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ21_HTML.gif
(3.3)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ22_HTML.gif
(3.4)
Suppose that ( λ , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq34_HTML.gif is a solution of (3.3), (3.4) with u = ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq85_HTML.gif. Then
| u ( t ) | p = λ p p 1 ( F ( ρ ) F ( u ( t ) ) ) , t [ 0 , 1 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equr_HTML.gif
and so
t ( p p 1 λ ) 1 / p = 0 u ( t ) ( F ( ρ ) F ( s ) ) 1 / p d s , t [ 0 , 1 2 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ23_HTML.gif
(3.5)
Putting t = 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq88_HTML.gif, we obtain
λ 1 / p = 2 ( p 1 p ) 1 / p 0 ρ ( F ( ρ ) F ( s ) ) 1 / p d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ24_HTML.gif
(3.6)

Hence λ (if exists) is uniquely determined by ρ.

If ρ S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq80_HTML.gif, we define λ ( ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq89_HTML.gif by (3.6) and u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq49_HTML.gif by (3.5). It is straightforward to verify that u is twice differentiable, u satisfies (3.3), (3.4), u > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq70_HTML.gif in ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq44_HTML.gif and u ( 1 / 2 ) = ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq90_HTML.gif. The continuity of λ ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq91_HTML.gif is implied by (3.6) and this completes the proof. □

Let
r = inf { u > 0 : f ( u ) = 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equs_HTML.gif

Then ( 0 , r ) S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq92_HTML.gif.

Lemma 3.2 Let (H1) and (H2′) hold, and let p ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq93_HTML.gif. Then
lim ρ 0 λ ( ρ ) = λ 1 f 0 , lim ρ r λ ( ρ ) = , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equt_HTML.gif

where λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq41_HTML.gif is the least eigenvalue of (1.9).

Proof We only deal with lim ρ 0 λ ( ρ ) = λ 1 f 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq94_HTML.gif. The other one can be treated by the same method.

To this end, we divide the proof into two cases.

Case 1. We show that f 0 = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq95_HTML.gif implies lim ρ 0 λ ( ρ ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq96_HTML.gif.

In this case, for any M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq97_HTML.gif, there is a positive number R such that f ( w ) > M w p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq98_HTML.gif for 0 w R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq99_HTML.gif. Thus, if ρ < R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq100_HTML.gif, then
F ( ρ ) F ( w ) = w ρ f ( v ) d v M p ( ρ p w p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equu_HTML.gif
for 0 w ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq101_HTML.gif. From (3.6), we have that for any ρ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq102_HTML.gif,
[ λ ( ρ ) ] 1 p = 2 ( p 1 p ) 1 p 0 ρ d w [ F ( ρ ) F ( w ) ] 1 / p 2 ( p 1 p ) 1 / p ( p M ) 1 / p 0 ρ d w [ ρ p w p ] 1 / p 2 ( p 1 M ) 1 / p 0 1 d w ρ [ 1 ( w ρ ) p ] 1 / p 2 ( p 1 M ) 1 / p ( 1 p 1 ) 1 / p 0 ( p 1 ) 1 / p d s [ 1 s p p 1 ] 1 / p 2 ( 1 M ) 1 / p 0 ( p 1 ) 1 / p d s [ 1 s p p 1 ] 1 / p = ( 1 M ) 1 / p π p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equv_HTML.gif
where
π p : = 2 0 ( p 1 ) 1 / p d s [ 1 s p p 1 ] 1 / p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equw_HTML.gif
see Zhang [16]. Hence
lim ρ 0 λ ( ρ ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equx_HTML.gif
Case 2. We show that f 0 = m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq103_HTML.gif for some m ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq104_HTML.gif implies that
lim ρ 0 λ ( ρ ) = p 1 p m τ p p = ( π p ) p f 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ25_HTML.gif
(3.7)
where
τ p = 2 0 1 [ p 1 v p ] 1 / p d v , p > 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ26_HTML.gif
(3.8)
In fact, (3.6) yields
[ m λ ( ρ ) ] 1 / p = 2 [ m ( p 1 ) p ] 1 p 0 ρ d w [ F ( ρ ) F ( w ) ] 1 / p = 2 ( p 1 p ) 1 / p 0 1 [ p 1 v p ] 1 / p d v 2 ( p 1 p ) 1 / p 0 1 [ p 1 v p ] 1 / p [ 1 + γ ( ρ , v ) ] 1 / p 1 [ 1 + γ ( ρ , v ) ] 1 / p d v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ27_HTML.gif
(3.9)
for p > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq105_HTML.gif, where
γ ( ρ , v ) = p m ρ v ρ [ f ( w ) m w p 1 ] d w ρ p ( 1 v p ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equy_HTML.gif

We will show that the last integral in (3.9) converges to zeros as ρ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq106_HTML.gif.

For 0 v 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq107_HTML.gif, using l’Hospital’s rule, it follows that as ρ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq106_HTML.gif,
| γ ( ρ , v ) | = p m ρ v ρ | f ( w ) m w p 1 | d w ρ p ( 1 v p ) p m 0 ρ | f ( w ) m w p 1 | d w ρ p ( 1 v p ) p m | f ( ρ ) m ρ p 1 | p ρ p 1 ( 1 v p ) = p m | f ( ρ ) ρ p 1 m | p ( 1 v p ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equz_HTML.gif
For 1 2 v 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq108_HTML.gif,
lim ρ 0 | γ ( ρ , v ) | = lim sup ρ 0 { | f ( w ) m w p 1 w p 1 | : 1 2 ρ w ρ } p m ρ p ( 1 v p ) ρ v ρ w p 1 d w = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equaa_HTML.gif
uniformly in v. Therefore, (3.9) implies
lim ρ 0 [ m λ ( ρ ) ] 1 / p = 2 ( p 1 p ) 1 p 0 1 [ p 1 v p ] 1 / p d v . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ28_HTML.gif
(3.10)

Therefore, (3.7) holds. □

From the definitions of α i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq109_HTML.gif and β i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq110_HTML.gif, we have that a i α i < β i a i + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq111_HTML.gif and ( α i , β i ) S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq112_HTML.gif for i = 1 , 2 , , n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq113_HTML.gif. Moreover, we have the following.

Lemma 3.3 Let p ( 1 , 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq114_HTML.gif. Then
  1. (i)

    lim ρ α i + λ ( ρ ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq115_HTML.gif;

     
  2. (ii)

    lim ρ β i λ ( ρ ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq116_HTML.gif.

     
Proof (i) Suppose firstly that f ( α i ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq117_HTML.gif. Since S is open, α i S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq118_HTML.gif and so there exists k : 0 < k < α i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq119_HTML.gif such that
F ( α i ) = F ( k ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equab_HTML.gif
Clearly k must be a local maximum for F and so f ( k ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq120_HTML.gif. If M = max { | f ( u ) | : 0 u b i } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq121_HTML.gif, then
f ( u ) M | u k | , 0 u b i . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equac_HTML.gif
Let
N = max { | f ( u ) | : 0 u b i } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equad_HTML.gif
Then if α i < ρ < b i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq122_HTML.gif,
F ( ρ ) F ( u ) = F ( ρ ) F ( α i ) + F ( k ) F ( u ) = ( ρ α i ) f ( ξ ) + ( k u ) f ( η ) , where  ξ ( α i , ρ )  and  η ( k , u ) N ( ρ α i ) + M ( k u ) 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equae_HTML.gif
Hence
( λ ( ρ ) ) 1 / p = 2 ( p 1 p ) 1 / p 0 ρ ( F ( ρ ) F ( s ) ) 1 / p d s 2 ( p 1 p ) 1 / p 0 α i ( N ( ρ α i ) + M ( k u ) 2 ) 1 / p d u = 0 α i H ρ ( u ) d u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equaf_HTML.gif
As ρ α i + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq123_HTML.gif, H ρ ( u ) = 2 ( p 1 p ) 1 / p ( N ( ρ α i ) + M ( k u ) 2 ) 1 / p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq124_HTML.gif is a nondecreasing sequence of measurable functions. Therefore, by the monotone convergence theorem and the assumption p 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq125_HTML.gif, it follows that
lim ρ α i + [ λ ( ρ ) ] 1 / p lim ρ α i + 0 α i H ρ ( u ) d u = 0 α i 2 ( p 1 p ) 1 / p M 1 / p [ k u ] 2 / p d u = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equag_HTML.gif

since k ( 0 , α i ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq126_HTML.gif.

Suppose next that f ( α i ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq127_HTML.gif. Then F ( α i ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq128_HTML.gif.

Since
F ( α i ) F ( u ) = f ( η ) ( α i u ) , where  η ( u , α i ) , | f ( u ) | = | f ( u ) f ( α i ) | M | u α i | . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equah_HTML.gif
Thus
0 α i [ F ( α i ) F ( u ) ] 1 / p d u 0 α i [ M | α i u | 2 ] 1 / p d u = 0 α i M 1 / p | α i u | 2 / p d u = . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equai_HTML.gif
  1. (ii)
    Let K 1 = max { | f ( u ) | : 0 u β i } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq129_HTML.gif and K 2 = max { | f ( u ) | : 0 u β i } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq130_HTML.gif. Since f ( β i ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq131_HTML.gif,
    f ( u ) K 2 | u β i | , 0 u < β i . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equ29_HTML.gif
    (3.11)
     
Hence, if 0 u ρ < β i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq132_HTML.gif, then it follows from (3.11) that
F ( ρ ) F ( u ) = F ( ρ ) F ( β i ) + F ( β i ) F ( u ) = ( ρ β i ) f ( ξ ) + ( β i u ) f ( η ) , where  ξ ( ρ , β i ) , η ( u , β i ) K 1 ( β i ρ ) + K 2 ( β i u ) 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equaj_HTML.gif
Hence, if 0 < ρ < β i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq133_HTML.gif,
( λ ( ρ ) ) 1 / p 2 ( p 1 p ) 1 / p 0 ρ ( K 1 ( β i ρ ) + K 2 ( β i u ) 2 ) 1 / p d u = 0 β i G ρ ( u ) d u , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equak_HTML.gif
where G ρ ( u ) = 2 ( p 1 p ) 1 / p ( K 1 ( β i ρ ) + K 2 ( β i u ) 2 ) 1 / p χ [ 0 , ρ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq134_HTML.gif and χ [ 0 , ρ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq135_HTML.gif denotes the characteristic function of [ 0 , ρ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq136_HTML.gif. As G ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq137_HTML.gif is a nondecreasing sequence of measurable functions, by the monotone convergence theorem
lim ρ β i [ λ ( ρ ) ] 1 / p lim ρ β i 0 β i G ρ ( u ) d u = 0 β i 2 ( p 1 p ) 1 / p K 2 1 / p | β i u | 2 / p d u = . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equal_HTML.gif

 □

Proof of Theorem 1.1 (a) follows from the continuity of ρ λ ( ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq86_HTML.gif and Lemma 3.2.
  1. (b)

    follows from the continuity of ρ λ ( ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq86_HTML.gif and Lemma 3.3.

     
  2. (c)
    ( λ , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq34_HTML.gif is any solution of (3.1), (3.2) if and only if
    u ( t ) = 0 t ( τ 1 / 2 λ f ( u ( s ) ) d s ) 1 p 1 d τ , t [ 0 , 1 2 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equam_HTML.gif
     
Hence
| u ( t ) | λ 1 p 1 0 1 / 2 ( τ 1 / 2 | f ( u ( s ) ) | d s ) 1 p 1 d τ λ 1 p 1 p 1 p ( 1 2 ) p p 1 ( sup y [ 0 , 1 ] | f ( u ( y ) ) | ) 1 p 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equan_HTML.gif
Now, if α i < u < β i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_IEq36_HTML.gif, then
α i λ 1 p 1 p 1 p ( 1 2 ) p p 1 ( sup s [ 0 , β i ] | f ( s ) | ) 1 p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equao_HTML.gif
and so
λ > ( α i C ) p 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-125/MediaObjects/13661_2012_Article_370_Equap_HTML.gif

 □

Declarations

Acknowledgements

The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC (No. 11061030), NSFC (No. 11126296), SRFDP (No. 20126203110004) and Gansu Provincial National Science Foundation of China (No. 1208RJZA258).

Authors’ Affiliations

(1)
Department of Mathematics, Northwest Normal University

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