Open Access

Existence of periodic solutions for a class of p-Laplacian equations

Boundary Value Problems20132013:96

DOI: 10.1186/1687-2770-2013-96

Received: 26 September 2012

Accepted: 5 April 2013

Published: 19 April 2013

Abstract

This paper is devoted to the existence of periodic solutions for the one-dimensional p-Laplacian equation

( ϕ p ( u ) ) = f ( t , u ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equa_HTML.gif

where ϕ p ( u ) = | u | p 2 u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq1_HTML.gif ( 1 < p < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq2_HTML.gif), f C ( [ 0 , 2 π ] × R , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq3_HTML.gif. By using some asymptotic interaction of the ratios f ( t , u ) | u | p 2 u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq4_HTML.gif and p 0 u f ( t , s ) d s | u | p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq5_HTML.gif with the Fučík spectrum of ( ϕ p ( u ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq6_HTML.gif related to periodic boundary condition, we establish a new existence theorem of periodic solutions for the one-dimensional p-Laplacian equation.

Keywords

periodic solutions p-Laplacian Fučík spectrum Leray-Schauder degree Borsuk theorem

1 Introduction and main results

In this paper, we are concerned with the existence of solutions for the following periodic boundary value problem:
{ ( ϕ p ( u ) ) = f ( t , u ) , u ( 0 ) = u ( 2 π ) , u ( 0 ) = u ( 2 π ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ1_HTML.gif
(1.1)

where ϕ p ( u ) = | u | p 2 u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq1_HTML.gif ( 1 < p < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq2_HTML.gif), f C ( [ 0 , 2 π ] × R , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq3_HTML.gif. A solution u of problem (1.1) means that u is C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq7_HTML.gif and ϕ p ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq8_HTML.gif is absolutely continuous such that (1.1) is satisfied for a.e. t [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq9_HTML.gif.

Existence and multiplicity of solutions of the periodic problems driven by the p-Laplacian have been obtained in the literature by many people (see [15]). Many solvability conditions for problem (1.1) were established by using the asymptotic interaction at infinity of the ratio f ( x , u ) | u | p 2 u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq10_HTML.gif with the Fučík spectrum for ( ϕ p ( u ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq11_HTML.gif under periodic boundary condition (see e.g., [2, 4, 69]). In [6], Del Pino, Manásevich and Murúa firstly defined the Fučík spectrum for ( ϕ p ( u ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq11_HTML.gif under periodic boundary value condition as the set Σ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq12_HTML.gif consisting of all the pairs ( λ + , λ ) R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq13_HTML.gif such that the equation
( ϕ p ( u ) ) = λ + ( u + ) p 1 λ ( u ) p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equb_HTML.gif
admits at least one nontrivial 2π-periodic solution (see [10] for p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq14_HTML.gif). Let
π p = 2 ( p 1 ) 1 p 0 1 ( 1 t p ) 1 p d t = 2 ( p 1 ) 1 p π p sin ( π p ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equc_HTML.gif
By [6], it follows that
Σ p = { ( λ + , λ ) R 2 : π p ( 1 λ + p + 1 λ p ) = 2 π k , k Z + } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equd_HTML.gif
Then they applied the Sturm’s comparison theorem and Leray-Schauder degree theory to prove that problem (1.1) is solvable if the following relations hold:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Eque_HTML.gif
uniformly for a.e. t [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq9_HTML.gif with p 1 , q 1 , p 2 , q 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq15_HTML.gif satisfying
2 π ( k + 1 ) π p < 1 p 2 p + 1 q 2 p 1 p 1 p + 1 q 1 p < 2 π k π p , k Z + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equf_HTML.gif
Clearly, in this case, we have ( [ p 1 , q 1 ] × [ p 2 , q 2 ] ) Σ p = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq16_HTML.gif, which is usually called that the nonlinearity f is nonresonant with respect to the Fučík spectrum Σ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq12_HTML.gif. In [11], Anane and Dakkak obtained a similar result by using the property of nodal set for eigenfunctions. If f is resonant with respect to Σ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq12_HTML.gif, i.e., there exists ( λ + , λ ) Σ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq17_HTML.gif such that lim u + f ( t , u ) | u | p 2 u = λ + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq18_HTML.gif, lim u f ( t , u ) | u | p 2 u = λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq19_HTML.gif uniformly for a.e. t [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq9_HTML.gif, together with the Landesman-Lazer type condition, Jiang [9] obtained the existence of solutions of (1.1) by applying the variational methods and symplectic transformations. In these works, either f is resonant or nonresonant with respect to Σ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq12_HTML.gif, the solvability of problem (1.1) was assured by assuming that the ratio f ( t , u ) | u | p 2 u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq4_HTML.gif stays at infinity in the pointwise sense asymptotically between two consecutive curves of Σ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq12_HTML.gif. Note that
lim inf s ± f ( t , s ) s lim inf s ± 2 F ( t , s ) s 2 lim sup s ± 2 F ( t , s ) s 2 lim sup s ± f ( t , s ) s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equg_HTML.gif
we can see that the conditions on the ratio 2 F ( t , s ) s 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq20_HTML.gif are more general than that on the ratio f ( t , s ) s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq21_HTML.gif. Recently, Liu and Li [2] studied the nondissipative p-Laplacian equation
( ϕ p ( u ) ) = c ( ϕ p ( u ) ) + g ( u ) p ( t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ2_HTML.gif
(1.2)
where c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq22_HTML.gif is a constant. Define G ( u ) = 0 u g ( s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq23_HTML.gif. They proved that (1.2) is solvable under the following assumptions:
  1. (1)

    There exist b , d 1 , d 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq24_HTML.gif such that d 1 g ( u ) | u | p 2 u d 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq25_HTML.gif for all | u | b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq26_HTML.gif;

     
  2. (2)

    lim u + p G ( u ) | u | p = λ + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq27_HTML.gif, lim u p G ( u ) | u | p = λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq28_HTML.gif with ( λ + , λ ) Σ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq29_HTML.gif.

     

Here, the potential function G is nonresonant with respect to Σ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq12_HTML.gif and the ratio g ( u ) | u | p 2 u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq30_HTML.gif is not required to stay at infinity in the pointwise sense asymptotically between two consecutive branches of Σ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq12_HTML.gif and it may even cross at infinity multiple Fučík spectrum curves.

In this paper, we want to obtain the solvability of problem (1.1) by using the asymptotic interaction at infinity of both the ratios f ( x , u ) | u | p 2 u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq10_HTML.gif and p F ( t , u ) | u | p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq31_HTML.gif with the Fučík spectrum for ( ϕ p ( u ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq11_HTML.gif under periodic boundary condition. Here, F ( t , u ) = 0 u f ( t , s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq32_HTML.gif. The goal is to obtain the existence of solutions of (1.1) by requiring neither the ratio f ( x , u ) | u | p 2 u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq10_HTML.gif stays at infinity in the pointwise sense asymptotically between two consecutive branches of Σ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq12_HTML.gif nor the limits lim u ± p F ( t , u ) | u | p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq33_HTML.gif exist. We shall prove that problem (1.1) admits a solution under the assumptions that the nonlinearity f has at most ( p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq34_HTML.gif-linear growth at infinity and the ratio f ( t , u ) | u | p 2 u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq35_HTML.gif has a L 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq36_HTML.gif limit as u ± https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq37_HTML.gif, while the ratio p F ( t , u ) | u | p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq31_HTML.gif stays at infinity in the pointwise sense asymptotically between two consecutive branches of Σ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq12_HTML.gif. Our result will complement the results in the literature on the solvability of problem (1.1) involving the Fučík spectrum.

For related works on resonant problems involving the Fučík spectrum, we also refer the interested readers to see [1219] and the references therein.

Our main result for problem (1.1) now reads as follows.

Theorem 1.1 Assume that f C ( [ 0 , 2 π ] × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq38_HTML.gif and the following conditions hold:
  1. (i)
    There exist constants C 1 , M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq39_HTML.gif such that
    | f ( t , u ) | C 1 ( 1 + | u | p 1 ) , a.e. t [ 0 , 2 π ] , | u | M ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ3_HTML.gif
    (1.3)
     
  2. (ii)
    There exists η ± L ( 0 , 2 π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq40_HTML.gif such that
    0 2 π | f ( t , u ) | u | p 2 u η ± ( t ) | d t 0 as u ± ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ4_HTML.gif
    (1.4)
     
  3. (iii)
    There exist constants p 1 , p 2 , q 1 , q 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq41_HTML.gif such that
    https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ5_HTML.gif
    (1.5)
     
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ6_HTML.gif
(1.6)
hold uniformly for a.e. t [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq9_HTML.gif with
( [ p 1 , p 2 ] × [ q 1 , q 2 ] ) Σ p = . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ7_HTML.gif
(1.7)

Then problem (1.1) admits a solution.

Remark If f ( t , u ) = a ( t ) | u | p 2 u + b ( t ) | u | p 2 u + e ( u ) + h ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq42_HTML.gif, where a , b , h C [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq43_HTML.gif with p 1 a ( t ) p 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq44_HTML.gif, q 1 b ( t ) q 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq45_HTML.gif and p 1 , p 2 , q 1 , q 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq41_HTML.gif satisfy (1.7), e is continuous on and lim u + e ( u ) | u | p 2 u = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq46_HTML.gif, then lim u + p F ( t , u ) | u | p = a ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq47_HTML.gif and lim u p F ( t , u ) | u | p = b ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq48_HTML.gif. By Theorem 1.1, it follows that problem (1.1) admits a solution. It is easily seen that the result of [16] cannot be applied to this case. Note that one can also obtain the solvability of (1.1) in this case by the result of [6], while in Theorem 1.1 we do not require the pointwise limit at infinity of the ratio f ( t , u ) | u | p 2 u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq4_HTML.gif as in [6].

For convenience, we introduce some notations and definitions. L p ( 0 , 2 π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq49_HTML.gif ( 1 < p < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq50_HTML.gif) denotes the usual Sobolev space with inner product , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq51_HTML.gif and norm p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq52_HTML.gif, respectively. C m [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq53_HTML.gif ( m N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq54_HTML.gif) denotes the space of m-times continuous differential real functions with norm
x C m = max t [ 0 , 2 π ] | x ( t ) | + max t [ 0 , 2 π ] | x ˙ ( t ) | + + max t [ 0 , 2 π ] | x ( m ) ( t ) | . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equh_HTML.gif

2 Proof of the main result

Denote by deg the Leray-Schauder degree. To prove Theorem 1.1, we need the following results.

Lemma 2.1 [20]

Let Ω be a bounded open region in a real Banach space X. Assume that K : Ω ¯ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq55_HTML.gif is completely continuous and p ( I K ) ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq56_HTML.gif. Then the equation ( I K ) ( x ) = p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq57_HTML.gif has a solution in Ω if deg ( I K , Ω , p ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq58_HTML.gif.

Lemma 2.2 (Borsuk Theorem [20])

Assume that X is a real Banach space. Let Ω be a symmetric bounded open region with θ Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq59_HTML.gif. Assume that K : Ω ¯ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq55_HTML.gif is completely continuous and odd with θ ( I K ) ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq60_HTML.gif. Then deg ( I K , Ω , θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq61_HTML.gif is odd.

Proof of Theorem 1.1 Take ( λ + , λ ) [ p 1 , p 2 ] × [ q 1 , q 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq62_HTML.gif. Consider the following homotopy problem:
{ ( ϕ p ( u ) ) = ( 1 μ ) ( λ + ( u + ) p 1 λ ( u ) p 1 ) + μ f ( t , u ) f μ ( t , u ) , u ( 0 ) = u ( 2 π ) , u ( 0 ) = u ( 2 π ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ8_HTML.gif
(2.1)

where μ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq63_HTML.gif.

By (1.3) and the regularity arguments, it follows that u C 1 [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq64_HTML.gif, and furthermore there exists a , b R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq65_HTML.gif such that, if u is a solution of problem (2.1), then
u C 1 a u + b . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ9_HTML.gif
(2.2)
In what follows, we shall prove that there exists C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq66_HTML.gif independent of μ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq63_HTML.gif such that u C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq67_HTML.gif for all possible solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq68_HTML.gif of (2.1). Assume by contradiction that there exist a sequence of number { μ n } [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq69_HTML.gif and corresponding solutions { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq70_HTML.gif of (2.1) such that
u n + as  n + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ10_HTML.gif
(2.3)
Set z n = u n u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq71_HTML.gif. Obviously, z n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq72_HTML.gif. Define
α n ( t ) = { f ( t , u n ) | u n | p 2 u n , u n ( t ) > M , 0 , u n ( t ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equi_HTML.gif
and
β n ( t ) = { f ( t , u n ) | u n | p 2 u n , u n ( t ) < M , 0 , u n ( t ) M . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equj_HTML.gif
By (1.3), there exists M 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq73_HTML.gif such that
| α n ( t ) | , | β n ( t ) | M 0 , a.e.  t [ 0 , 2 π ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equk_HTML.gif
Then there exist α 0 , β 0 L ( 0 , 2 π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq74_HTML.gif such that
α n ( t ) α 0 ( t ) , β n ( t ) β 0 ( t ) in  L ( 0 , 2 π ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ11_HTML.gif
(2.4)
In addition, using (1.3) and the regularity arguments, there exists M 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq75_HTML.gif such that, for each n, we have z n C 1 M 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq76_HTML.gif, and thus there exists z 0 C 1 [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq77_HTML.gif such that, passing to a subsequence if possible,
z n z 0 in  C 1 [ 0 , 2 π ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ12_HTML.gif
(2.5)
Clearly, z 0 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq78_HTML.gif. In view of { μ n } [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq69_HTML.gif, there exists μ 0 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq79_HTML.gif such that, passing to a subsequence if possible,
μ n μ 0 as  n + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ13_HTML.gif
(2.6)
Note that for μ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq80_HTML.gif, problem (2.1) has only the trivial solution, it follows that μ 0 ( 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq81_HTML.gif. Denote α ¯ ( t ) = ( 1 μ 0 ) λ + + μ 0 α 0 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq82_HTML.gif, β ¯ ( t ) = ( 1 μ 0 ) λ + μ 0 β 0 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq83_HTML.gif. It is easily seen that z 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq84_HTML.gif is a nontrivial solution of the following problem:
{ ( ϕ p ( z 0 ) ) = α ¯ ( t ) ( z 0 + ) p 1 β ¯ ( t ) ( z 0 ) p 1 , z 0 ( 0 ) = z 0 ( 2 π ) , z 0 ( 0 ) = z 0 ( 2 π ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ14_HTML.gif
(2.7)
We now distinguish three cases:
  1. (i)

    z 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq84_HTML.gif changes sign in [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq85_HTML.gif;

     
  2. (ii)

    z 0 ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq86_HTML.gif, t [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq87_HTML.gif;

     
  3. (iii)

    z 0 ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq88_HTML.gif, t [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq89_HTML.gif.

     

In the following, it will be shown that each case leads to a contradiction.

Case (i). Let
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equl_HTML.gif
Then, as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq90_HTML.gif, we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equm_HTML.gif
In addition, as shown in [11], we have | I 0 | = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq91_HTML.gif. Define
η + ( t ) = { η ( t ) , t I + , α 0 ( t ) , t I , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equn_HTML.gif
and
η ( t ) = { β 0 ( t ) , t I + , η ( t ) , t I . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equo_HTML.gif
By (1.4) and (2.4), it follows that
α 0 ( t ) η + ( t ) , β 0 ( t ) η ( t ) , a.e.  t [ 0 , 2 π ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equp_HTML.gif
Thus, z 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq84_HTML.gif satisfies
{ ( ϕ p ( z 0 ) ) = α ˜ ( t ) ( z 0 + ) p 1 β ˜ ( t ) ( z 0 ) p 1 , z 0 ( 0 ) = z 0 ( 2 π ) , z 0 ( 0 ) = z 0 ( 2 π ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ15_HTML.gif
(2.8)

Here, α ˜ ( t ) = ( 1 μ 0 ) λ + + μ 0 η + ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq92_HTML.gif, β ˜ ( t ) = ( 1 μ 0 ) λ + μ 0 η ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq93_HTML.gif.

Now we prove that there exist n ¯ Z + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq94_HTML.gif and 0 < κ 1 < 1 < κ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq95_HTML.gif such that
κ 1 max u n min u n κ 2 , n n ¯ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ16_HTML.gif
(2.9)
In fact, if not, we assume, by contradiction, that there exists a subsequence of { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq70_HTML.gif, we still denote it as { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq70_HTML.gif with max u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq96_HTML.gif and min u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq97_HTML.gif, such that
max u n min u n 0 or max u n min u n + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equq_HTML.gif
Combing with (2.5), z 0 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq78_HTML.gif and the fact that z 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq84_HTML.gif changes sign, we obtain
max u n u n / ( min u n u n ) max z 0 min z 0 > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equr_HTML.gif

A contradiction. Hence, (2.9) holds.

For any ( t , μ ) [ 0 , 2 π ] × [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq98_HTML.gif, define
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equs_HTML.gif
and
F ¯ 1 ( t , s , μ ) = 0 s f ¯ 1 ( t , τ , μ ) d τ , F ¯ 2 ( t , r , μ ) = 0 r f ¯ 2 ( t , τ , μ ) d τ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equt_HTML.gif
Denote s n = max u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq99_HTML.gif, r n = min u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq100_HTML.gif. Then by (2.9) it follows that s n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq101_HTML.gif and r n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq102_HTML.gif. Taking t n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq103_HTML.gif such that u n ( t n ) = s n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq104_HTML.gif, t n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq105_HTML.gif is the nearest point satisfying t n 0 < t n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq106_HTML.gif and u n ( t n 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq107_HTML.gif. Since t n 0 , t n [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq108_HTML.gif, there exist t ¯ 0 , t ¯ [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq109_HTML.gif such that
t n 0 t ¯ 0 , t n t ¯ as  n + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ17_HTML.gif
(2.10)
By (2.5), we obtain z 0 ( t ¯ 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq110_HTML.gif, z 0 ( t ¯ ) = max t [ 0 , 2 π ] z 0 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq111_HTML.gif. Note that u n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq112_HTML.gif, we have u n ( t ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq113_HTML.gif, t ( t ¯ 0 , t ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq114_HTML.gif. Hence, together with μ n μ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq115_HTML.gif and (1.4), there exist subsequences of { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq70_HTML.gif and { μ n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq116_HTML.gif, we still denote them as { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq70_HTML.gif and { μ n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq116_HTML.gif, such that, for a.e. t [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq9_HTML.gif,
f ¯ 1 ( t , u n ( τ ) , μ n ) ( u n ( τ ) ) p 1 0 , a.e.  τ ( t ¯ 0 , t ¯ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equu_HTML.gif
Using (1.3), for a.e. t [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq9_HTML.gif, { f ¯ 1 ( t , u n ( τ ) , μ n ) ( u n ( τ ) ) p 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq117_HTML.gif is uniformly bounded with respect to τ ( t ¯ 0 , t ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq118_HTML.gif, we obtain by the Lebesgue dominated convergence theorem that
t ¯ 0 t ¯ | f ¯ 1 ( t , u n ( τ ) , μ n ) ( u n ( τ ) ) p 1 | d τ 0 , uniformly for a.e.  t [ 0 , 2 π ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equv_HTML.gif
Thus,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ18_HTML.gif
(2.11)
By (1.4) and (2.2), we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equw_HTML.gif
In view of (2.11), we obtain that
| p F ¯ 1 ( t , s n , μ n ) s n p | 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ19_HTML.gif
(2.12)
holds uniformly for a.e. t [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq9_HTML.gif. Similarly,
| p F ¯ 2 ( t , r n , μ n ) | r n | p | 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ20_HTML.gif
(2.13)

holds uniformly for a.e. t [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq9_HTML.gif.

On the other hand, for { s n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq119_HTML.gif, { r n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq120_HTML.gif satisfying (2.12)-(2.13), denoting
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equx_HTML.gif
we obtain by (1.5)-(1.6) that
p 1 ξ 1 ( t ) q 1 , p 2 ξ 2 ( t ) q 2 , a.e.  t [ 0 , 2 π ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ21_HTML.gif
(2.14)
Using μ n μ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq115_HTML.gif, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ22_HTML.gif
(2.15)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ23_HTML.gif
(2.16)
We claim that there exists subinterval I 1 [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq121_HTML.gif with | I 1 | > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq122_HTML.gif such that
ξ 1 ( t ) η + ( t ) 0 , t I 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ24_HTML.gif
(2.17)
or subinterval I 2 [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq123_HTML.gif with | I 2 | > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq124_HTML.gif such that
ξ 2 ( t ) η ( t ) 0 , t I 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ25_HTML.gif
(2.18)
Indeed, if not, we assume that η + ( t ) = ξ 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq125_HTML.gif, η ( t ) = ξ 2 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq126_HTML.gif, a.e. t [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq9_HTML.gif. Together with the choosing of λ + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq127_HTML.gif, λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq128_HTML.gif and (2.14), we get
p 1 α ˜ ( t ) q 1 , p 2 β ˜ ( t ) q 2 , a.e.  t [ 0 , 2 π ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equy_HTML.gif

Then by (1.7), it follows that z 0 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq129_HTML.gif. A contradiction. Combining (2.12)-(2.13) with (2.15)-(2.18), we obtain a contradiction.

Case (ii). In this case, we have
s n +  and  { r n }  is uniformly bounded . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equz_HTML.gif
Using similar arguments as in Case (i), by (1.4) and (2.4) it follows that α 0 ( t ) η + ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq130_HTML.gif, t [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq87_HTML.gif. Taking f ¯ + = f ¯ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq131_HTML.gif, F ¯ + = F ¯ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq132_HTML.gif, a.e. t [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq9_HTML.gif. We can see that there exists subsequence of  { s n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq119_HTML.gif, which is still denoted by { s n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq119_HTML.gif, such that
| p F ¯ + ( t , s n , μ n ) s n p | 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ26_HTML.gif
(2.19)
holds uniformly for a.e. t [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq9_HTML.gif. On the other hand, for { s n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq119_HTML.gif satisfying (2.19), denoting
ξ + ( t ) = lim inf n + p F ( t , s n ) | s n | p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equaa_HTML.gif
we obtain by (1.5) that
p 1 ξ + ( t ) q 1 , a.e.  t [ 0 , 2 π ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ27_HTML.gif
(2.20)
Using μ n μ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq115_HTML.gif, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ28_HTML.gif
(2.21)
We shall show that there exists subinterval I + [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq133_HTML.gif with | I + | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq134_HTML.gif such that
ξ + ( t ) η + ( t ) 0 , t I + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ29_HTML.gif
(2.22)
In fact, if not, we assume that η + ( t ) = ξ + ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq135_HTML.gif, a.e. t [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq136_HTML.gif. By the choosing of λ + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq127_HTML.gif and (2.20), we get p 1 α ˜ ( t ) q 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq137_HTML.gif, a.e. t [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq9_HTML.gif. Thus, z 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq84_HTML.gif is a nontrivial solution of the following problem:
{ ( ϕ p ( z 0 ) ) = α ˜ ( t ) z 0 p 1 , z 0 ( 0 ) = z 0 ( 2 π ) , z 0 ( 0 ) = z 0 ( 2 π ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ30_HTML.gif
(2.23)
Taking 1 as test function in problem (2.23), we get
0 = 0 2 π α ˜ ( t ) z 0 p 1 d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ31_HTML.gif
(2.24)

By α ˜ ( t ) p 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq138_HTML.gif for a.e. t [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq9_HTML.gif, it follows that z 0 ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq139_HTML.gif for a.e. t [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq9_HTML.gif, which is contrary to that z 0 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq78_HTML.gif. Hence, (2.22) holds. Clearly, (2.21)-(2.22) contradict (2.19).

Case (iii). In this case, r n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq102_HTML.gif and { s n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq119_HTML.gif is uniformly bounded. Similar arguments as in Case (ii) imply a contradiction.

In a word, (2.3) cannot hold, and hence by (2.2) there exists C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq66_HTML.gif independent of μ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq63_HTML.gif such that, if u is a solution of problem (2.1), then
u C 1 C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ32_HTML.gif
(2.25)
Note that, for each h L ( 0 , 2 π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq140_HTML.gif, the problem
{ ( ϕ p ( u ) ) + ϕ p ( u ) = h ( t ) , u ( 0 ) = u ( 2 π ) , u ( 0 ) = u ( 2 π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equ33_HTML.gif
(2.26)
has a unique solution G p ( h ) C 1 [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq141_HTML.gif. Clearly, the operator G p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq142_HTML.gif seen as an operator from C [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq143_HTML.gif into C 1 [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq144_HTML.gif is completely continuous. Define ψ : C 1 [ 0 , 2 π ] C [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq145_HTML.gif by ψ ( u ) ( t ) = f ( t , u ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq146_HTML.gif. Then solving problem (1.1) is equivalent to finding solutions in C 1 [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq144_HTML.gif of the equation
u G p ( ψ ( u ) ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equab_HTML.gif
Let ( α , β ) [ p 1 , q 1 ] × [ p 1 , q 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq147_HTML.gif. Define the operator T α , β : C 1 [ 0 , 2 π ] C 1 [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq148_HTML.gif by T α , β ( u ) = G p ( ϕ p ( u ) + α ( u + ) p 1 β ( u ) p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq149_HTML.gif. Denote B R = { u C 1 [ 0 , 2 π ] : u C 1 < R , R R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq150_HTML.gif. Clearly, deg ( I T α , β , B R , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq151_HTML.gif is well defined for all R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq152_HTML.gif. Owing to λ + λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq153_HTML.gif, there is a continuous curve α ( τ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq154_HTML.gif, β ( τ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq155_HTML.gif, τ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq156_HTML.gif, whose image is in R 2 Σ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq157_HTML.gif and ( λ , λ ) R Σ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq158_HTML.gif such that ( α ( 0 ) , β ( 0 ) ) = ( λ + , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq159_HTML.gif, ( α ( 1 ) , β ( 1 ) ) = ( λ , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq160_HTML.gif. From the invariance property of Leray-Schauder degree under compact homotopies, it follows that the degree deg ( I T α ( τ ) , β ( τ ) , B R , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq161_HTML.gif is constant for τ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq156_HTML.gif. Obviously, the operator T λ , λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq162_HTML.gif is odd. By the Borsuk’s theorem, it follows that deg ( I T λ , λ , B R , 0 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq163_HTML.gif for all R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq152_HTML.gif. Thus,
deg ( I T λ + , λ , B R , 0 ) 0 , R > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equac_HTML.gif
Consider the following homotopy:
H ( μ , u ) = G p ( ϕ p ( u ) + ( 1 μ ) ( λ + ( u + ) p 1 λ ( u ) p 1 ) + μ ψ ( u ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equad_HTML.gif
for ( μ , u ) [ 0 , 1 ] × C 1 [ 0 , 2 π ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq164_HTML.gif. By (2.25), we can see that there exists R 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_IEq165_HTML.gif such that
H ( μ , u ) u , μ [ 0 , 1 ] , u B R 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equae_HTML.gif
From the invariance property of Leray-Schauder degree, it follows that
deg ( I H ( 1 , ) , B R 0 , 0 ) = deg ( I H ( 0 , ) , B R 0 , 0 ) = deg ( I T λ + , λ , B R 0 , 0 ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-96/MediaObjects/13661_2012_Article_351_Equaf_HTML.gif

Hence, problem (1.1) has a solution. The proof is complete. □

Declarations

Acknowledgements

The first author sincerely thanks Professor Yong Li and Doctor Yixian Gao for their many useful suggestions and the both authors thank Professor Zhi-Qiang Wang for many helpful discussions and his invitation to Chern Institute of Mathematics. The first author is partially supported by the NSFC Grant (11101178), NSFJP Grant (201215184) and FSIIP of Jilin University (201103203). The second author is partially sup- ported by NSFC Grant (11226123).

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Northeast Normal University
(2)
College of Mathematics, Jilin University
(3)
College of Mathematics and Information Science, Shaanxi Normal University

References

  1. Aizicovici S, Papageorgiou NS, Staicu V: Nonlinear resonant periodic problems with concave terms. J. Math. Anal. Appl. 2011, 375: 342-364. 10.1016/j.jmaa.2010.09.009MathSciNetView Article
  2. Liu W, Li Y: Existence of periodic solutions for p -Laplacian equation under the frame of Fučík spectrum. Acta Math. Sin. Engl. Ser. 2011, 27: 545-554. 10.1007/s10114-011-9719-1MathSciNetView Article
  3. Manásevich R, Mawhin J: Periodic solutions for nonlinear systems with p -Laplacian-like operators. J. Differ. Equ. 1998, 145: 367-393. 10.1006/jdeq.1998.3425View Article
  4. Reichel W, Walter W: Sturm-Liouville type problems for the p -Laplacian under asymptotic nonresonance conditions. J. Differ. Equ. 1999, 156: 50-70. 10.1006/jdeq.1998.3611MathSciNetView Article
  5. Yang X, Kim Y, Lo K: Periodic solutions for a generalized p -Laplacian equation. Appl. Math. Lett. 2012, 25: 586-589. 10.1016/j.aml.2011.09.064MathSciNetView Article
  6. Del Pino M, Manásevich R, Murúa A: Existence and multiplicity of solutions with prescribed period for a second order quasilinear ODE. Nonlinear Anal. 1992, 18: 79-92. 10.1016/0362-546X(92)90048-JMathSciNetView Article
  7. Fabry C, Fayyad D: Periodic solutions of second order differential equations with a p -Laplacian and asymmetric nonlinearities. Rend. Ist. Mat. Univ. Trieste 1992, 24: 207-227.MathSciNet
  8. Fabry C, Manásevich R: Equations with a p -Laplacian and an asymmetric nonlinear term. Discrete Contin. Dyn. Syst. 2001, 7: 545-557.View Article
  9. Jiang M: A Landesman-Lazer type theorem for periodic solutions of the resonant asymmetric p -Laplacian equation. Acta Math. Sin. 2005, 21: 1219-1228. 10.1007/s10114-004-0459-3View Article
  10. Drábek P Pitman Research Notes in Mathematics 264. Solvability and Bifurcations of Nonlinear Equations 1992.
  11. Anane A, Dakkak A: Nonexistence of nontrivial solutions for an asymmetric problem with weights. Proyecciones 2000, 19: 43-52.MathSciNet
  12. Boccardo L, Drábek P, Giachetti D, Kućera M: Generalization of Fredholm alternative for nonlinear differential operators. Nonlinear Anal. 1986, 10: 1083-1103. 10.1016/0362-546X(86)90091-XMathSciNetView Article
  13. Drábek P, Invernizzi S: On the periodic boundary value problem for forced Duffing equation with jumping nonlinearity. Nonlinear Anal. 1986, 10: 643-650. 10.1016/0362-546X(86)90124-0MathSciNetView Article
  14. Fonda A: On the existence of periodic solutions for scalar second order differential equations when only the asymptotic behaviour of the potential is known. Proc. Am. Math. Soc. 1993, 119: 439-445. 10.1090/S0002-9939-1993-1154246-4MathSciNetView Article
  15. Habets P, Omari P, Zanolin F: Nonresonance conditions on the potential with respect to the Fučík spectrum for the periodic boundary value problem. Rocky Mt. J. Math. 1995, 25: 1305-1340. 10.1216/rmjm/1181072148MathSciNetView Article
  16. Liu W, Li Y: Existence of 2 π -periodic solutions for the non-dissipative Duffing equation under asymptotic behaviors of potential function. Z. Angew. Math. Phys. 2006, 57: 1-11.MathSciNetView Article
  17. Omari P, Zanolin F: Nonresonance conditions on the potential for a second-order periodic boundary value problem. Proc. Am. Math. Soc. 1993, 117: 125-135. 10.1090/S0002-9939-1993-1143021-2MathSciNetView Article
  18. Zhang M: Nonresonance conditions for asymptotically positively homogeneous differential systems: the Fučík spectrum and its generalization. J. Differ. Equ. 1998, 145: 332-366. 10.1006/jdeq.1997.3403View Article
  19. Zhang M: The rotation number approach to the periodic Fučík spectrum. J. Differ. Equ. 2002, 185: 74-96. 10.1006/jdeq.2002.4168View Article
  20. Deimling K: Nonlinear Functional Analysis. Springer, New York; 1985.View Article

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© Chang and Qiao; licensee Springer 2013

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