Open Access

Existence and multiplicity of solutions for nonlocal p(x)-Laplacian equations with nonlinear Neumann boundary conditions

Boundary Value Problems20122012:1

DOI: 10.1186/1687-2770-2012-1

Received: 29 August 2011

Accepted: 4 January 2012

Published: 4 January 2012

Abstract

In this article, we study the nonlocal p(x)-Laplacian problem of the following form

a ( Ω 1 p ( x ) ( | u | p ( x ) + | u | p ( x ) ) d x ) ( - d i v ( | u | p ( x ) - 2 u ) + | u | p ( x ) - 2 u ) = b ( Ω F ( x , u ) d x ) f ( x , u ) in Ω a Ω 1 p ( x ) ( | u | p ( x ) + | u | p ( x ) ) d x | u | p ( x ) - 2 u ν = g ( x , u ) on Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equa_HTML.gif

where Ω is a smooth bounded domain and ν is the outward normal vector on the boundary ∂Ω, and F ( x , u ) = 0 u f ( x , t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq1_HTML.gif. By using the variational method and the theory of the variable exponent Sobolev space, under appropriate assumptions on f, g, a and b, we obtain some results on existence and multiplicity of solutions of the problem.

Mathematics Subject Classification (2000): 35B38; 35D05; 35J20.

Keywords

critical points p(x)-Laplacian nonlocal problem variable exponent Sobolev spaces nonlinear Neumann boundary conditions

1 Introduction

In this article, we consider the following problem
( P ) a Ω 1 p ( x ) ( | u | p ( x ) + | u | p ( x ) ) d x ( - d i v ( | u | p ( x ) - 2 u ) + | u | p ( x ) - 2 u ) = b Ω F ( x , u ) d x f ( x , u ) in Ω a Ω 1 p ( x ) ( | u | p ( x ) + | u | p ( x ) ) d x | u | p ( x ) - 2 u ν = g ( x , u ) on Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equb_HTML.gif

where Ω is a smooth bounded domain in R N , p C Ω ̄ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq2_HTML.gif with 1 < p- := infΩ p(x) ≤ p(x) ≤ p+ := supΩ p(x) < N, a(t) is a continuous real-valued function, f : Ω × RR, g : ∂Ω × RR satisfy the Caratheodory condition, and F ( x , u ) = 0 u f ( x , t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq3_HTML.gif. Since the equation contains an integral related to the unknown u over Ω, it is no longer an identity pointwise, and therefore is often called nonlocal problem.

Kirchhoff [1] has investigated an equation
ρ 2 u t 2 - P 0 h + E 2 L 0 L u x 2 d x 2 u x 2 = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equc_HTML.gif
which is called the Kirchhoff equation. Various equations of Kirchhoff type have been studied by many authors, especially after the work of Lions [2], where a functional analysis framework for the problem was proposed; see e.g. [36] for some interesting results and further references. In the following, a key work on nonlocal elliptic problems is the article by Chipot and Rodrigues [7]. They studied nonlocal boundary value problems and unilateral problems with several applications. And now the study of nonlocal elliptic problem has already been extended to the case involving the p-Laplacian; see e.g. [8, 9]. Recently, Autuori, Pucci and Salvatori [10] have investigated the Kirchhoff type equation involving the p(x)-Laplacian of the form
u t t - M Ω 1 p ( x ) | u | p ( x ) d x Δ p ( x ) u + Q ( t , x , u , u t ) + f ( x , u ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equd_HTML.gif

The study of the stationary version of Kirchhoff type problems has received considerable attention in recent years; see e.g. [5, 1116].

The operator Δp(x)u = div(|u|p(x)-2u) is called p(x)-Laplacian, which becomes p-Laplacian when p(x) ≡ p (a constant). The p(x)-Laplacian possesses more complicated nonlinearities than p-Laplacian. The study of various mathematical problems with variable exponent are interesting in applications and raise many difficult mathematical problems. We refer the readers to [1723] for the study of p(x)-Laplacian equations and the corresponding variational problems.

Corrêa and Figueiredo [13] presented several sufficient conditions for the existence of positive solutions to a class of nonlocal boundary value problems of the p-Kirchhoff type equation. Fan and Zhang [20] studied p(x)-Laplacian equation with the nonlinearity f satisfying Ambrosetti-Rabinowitz condition. The p(x)-Kirchhoff type equations with Dirichlet boundary value problems have been studied by Dai and Hao [24], and much weaker conditions have been given by Fan [25]. The elliptic problems with nonlinear boundary conditions have attracted expensive interest in recent years, for example, for the Laplacian with nonlinear boundary conditions see [2630], for elliptic systems with nonlinear boundary conditions see [31, 32], for the p-Laplacian with nonlinear boundary conditions of different type see [3337], and for the p(x)-Laplacian with nonlinear boundary conditions see [3840]. Motivated by above, we focus the case of nonlocal p(x)-Laplacian problems with nonlinear Neumann boundary conditions. This is a new topics even when p(x) ≡ p is a constant.

This rest of the article is organized as follows. In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces. In Section 3, we consider the case where the energy functional associated with problem (P) is coercive. And in Section 4, we consider the case where the energy functional possesses the Mountain Pass geometry.

2 Preliminaries

In order to discuss problem (P), we need some theories on variable exponent Sobolev space W1,p(x)(Ω). For ease of exposition we state some basic properties of space W1,p(x)(Ω) (for details, see [22, 41, 42]).

Let Ω be a bounded domain of R N , denote
C + ( Ω ̄ ) = { p | p C ( Ω ̄ ) , p ( x ) > 1 , x Ω ̄ } , p + = max x Ω ̄ p ( x ) , p - = min x Ω ̄ p ( x ) , p C ( Ω ̄ ) , L p ( x ) ( Ω ) = u | u is a measurable real - valued function on Ω , Ω | u | p ( x ) d x < , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Eque_HTML.gif
we can introduce the norm on Lp(x)(Ω) by
u p ( x ) = inf λ > 0 : Ω u ( x ) λ p ( x ) d x 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equf_HTML.gif

and (Lp(x)(Ω), | · |p(x)) becomes a Banach space, we call it the variable exponent Lebesgue space.

The space W1, p(x)(Ω) is defined by
W 1 , p ( x ) ( Ω ) = { u L p ( x ) ( Ω ) | | u | L p ( x ) ( Ω ) } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equg_HTML.gif
and it can be equipped with the norm
| | u | | = | u | p ( x ) + | u | p ( x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equh_HTML.gif

where |u|p(x)= ||u||p(x); and we denote by W 0 1 , p ( x ) ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq4_HTML.gif the closure of C 0 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq5_HTML.gif in W1, p(x)(Ω), p * = N p ( x ) N - p ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq6_HTML.gif, p * = ( N - 1 ) p ( x ) N - p ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq7_HTML.gif, when p(x) < N, and p* = p* = ∞, when p(x) > N.

Proposition 2.1 [22, 41]. (1) If p C + ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq8_HTML.gif, the space (Lp(x)(Ω), | · |p(x)) is a separable, uniform convex Banach space, and its dual space is Lq(x)(Ω), where 1/q(x) + 1/p(x) = 1. For any u Lp(x)(Ω) and v Lq(x)(Ω), we have
Ω u v d x ( 1 p - + 1 q - ) | u | p ( x ) | v | q ( x ) ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equi_HTML.gif

(2) If p 1 , p 2 C + ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq9_HTML.gif, p1 (x) ≤ p2 (x), for any x Ω, then L p 2 ( x ) ( Ω ) L p 1 ( x ) ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq10_HTML.gif, and the imbedding is continuous.

Proposition 2.2 [22]. If f : Ω × RR is a Caratheodory function and satisfies
| f ( x , s ) | d ( x ) + e | s | p 1 ( x ) p 2 ( x ) , f o r a n y x Ω , s R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equj_HTML.gif

where p 1 , p 2 C + ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq11_HTML.gif, d L p 2 ( x ) ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq12_HTML.gif, d(x) ≥ 0 and e ≥ 0 is a constant, then the superposition operator from L p 1 ( x ) ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq13_HTML.gif to L p 2 ( x ) ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq14_HTML.gif defined by (N f (u)) (x) = f (x, u (x)) is a continuous and bounded operator.

Proposition 2.3 [22]. If we denote
ρ ( u ) = Ω | u | p ( x ) d x , u L p ( x ) ( Ω ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equk_HTML.gif
then for u, u n Lp(x)(Ω)
  1. (1)

    |u (x)|p(x)< 1(= 1; > 1) ρ (u) < 1(= 1; > 1);

     
  2. (2)

    | u ( x ) | p ( x ) > 1 | u | p ( x ) p - ρ ( u ) | u | p ( x ) p + ; | u ( x ) | p ( x ) < 1 | u | p ( x ) p - ρ ( u ) | u | p ( x ) p + ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq15_HTML.gif

     
  3. (3)

    | u n ( x ) | p ( x ) 0 ρ ( u n ) 0 as n ; | u n ( x ) | p ( x ) ρ ( u n ) as n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq16_HTML.gif

     
Proposition 2.4 [22]. If u, u n Lp(x)(Ω), n = 1, 2, ..., then the following statements are equivalent to each other
  1. (1)

    limk → ∞|u k - u|p(x)= 0;

     
  2. (2)

    limk → ∞ρ |u k - u| = 0;

     
  3. (3)

    u k u in measure in Ω and limk → ∞ρ (u k ) = ρ (u).

     

Proposition 2.5 [22]. (1) If p C + ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq17_HTML.gif, then W 0 1 , p ( x ) ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq18_HTML.gif and W1,p(x)(Ω) are separable reflexive Banach spaces;

(2) if q C + ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq19_HTML.gif and q (x) < p* (x) for any x Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq20_HTML.gif, then the imbedding from W1, p(x)(Ω) to Lq(x)(Ω) is compact and continuous;

(3) if q C + ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq19_HTML.gif and q (x) < p* (x) for any x Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq20_HTML.gif, then the trace imbedding from W1, p(x)(Ω) to Lq(x)(∂Ω)is compact and continuous;

(4) (Poincare inequality) There is a constant C > 0, such that
| u | p ( x ) C | u | p ( x ) u W 0 1 , p ( x ) ( Ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equl_HTML.gif

So, |u|p(x)is a norm equivalent to the norm || u || in the space W 0 1 , p ( x ) ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq21_HTML.gif.

3 Coercive functionals

In this and the next sections we consider the nonlocal p(x)-Laplacian-Neumann problem (P), where a and b are two real functions satisfying the following conditions

(a1) a : (0, + ∞) → (0, + ∞) is continuous and a L1 (0, t) for any t > 0.

(b1) b : RR is continuous.

Notice that the function a satisfies (a1) may be singular at t = 0. And f, g satisfying

(fl) f : Ω × RR satisfies the Caratheodory condition and there exist two constants C1 ≥ 0, C2 ≥ 0 such that
| f ( x , t ) | C 1 + C 2 | t | q 1 ( x ) - 1 , ( x , t ) Ω × R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equm_HTML.gif

where q 1 C + ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq22_HTML.gif and q1 (x) < p* (x), x Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq23_HTML.gif.

(g1) g : ∂Ω × RR satisfies the Caratheodory condition and there exist two constants C 1 0 , C 2 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq24_HTML.gif such that
| g ( x , t ) | C 1 + C 2 | t | q 2 ( x ) - 1 , ( x , t ) Ω × R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equn_HTML.gif

where q2 C+ (∂Ω) and q2 (x) < p* (x), x ∂Ω. For simplicity we write X = W1, p(x)(Ω), denote by C the general positive constant (the exact value may change from line to line).

Define

a ^ ( t ) = 0 t a ( s ) d s , b ^ ( t ) = 0 t b ( s ) d s , t R , I 1 ( u ) = Ω 1 p ( x ) | u | p ( x ) + | u | p ( x ) d x , I 2 ( u ) = Ω F ( x , u ) d x , u X , J ( u ) = a ^ ( I 1 ( u ) ) = a ^ Ω 1 p ( x ) | u | p ( x ) + | u | p ( x ) d x , Φ ( u ) = b ^ ( I 2 ( u ) ) = b ^ Ω F ( x , u ) d x and Ψ ( u ) = Ω G ( x , u ) d σ , u X , E ( u ) = J ( u ) - Φ ( u ) - Ψ ( u ) , u X , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq25_HTML.gif,

where F ( x , u ) = 0 u f ( x , t ) d t , G ( x , u ) = 0 u g ( x , t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq26_HTML.gif.

Lemma 3.1. Let (f1), (g1) (a1) and (b1) hold. Then the following statements hold true:

(1) a ^ C 0 ( [ 0 , ) ) C 1 ( ( 0 , ) ) , a ^ ( 0 ) = 0 , a ^ ( t ) = a ( t ) > 0 ; b ^ C 1 ( R ) , b ^ ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq27_HTML.gif.

(2) J, Φ, Ψ and E C0 (X), J (0) = Φ (0) = Ψ (0) = E (0) = 0. Furthermore J C1 (X\{0}), Φ, Ψ C1 (X), E C1 (X\{0}). And for every u X\{0}, v X, we have
E ( u ) v = a Ω 1 p ( x ) | u | p ( x ) + | u | p ( x ) d x Ω | u | p ( x ) - 2 u v + | u | p ( x ) - 2 u v d x - b Ω F ( x , u ) d x Ω f ( x , u ) v d x - Ω g ( x , u ) v d σ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equo_HTML.gif

Thus u X\{0} is a (weak) solution of (P) if and only if u is a critical point of E.

(3) The functional J : XR is sequentially weakly lower semi-continuous, Φ, Ψ: XR are sequentially weakly continuous, and thus E is sequentially weakly lower semi-continuous.

(4) The mappings Φ' and Ψ' are sequentially weakly-strongly continuous, namely, u n u in X implies Φ' (u n ) → Φ' (u) in X*. For any open set D X\{0} with D ¯ X \ { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq28_HTML.gif, The mappings J' and E : D ¯ X * https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq29_HTML.gif are bounded, and are of type (S+), namely,
u n u and lim ¯ n J ( u n ) ( u n u ) 0 , implies u n u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equp_HTML.gif

Definition 3.1. Let c R, a C1-functional E : XR satisfies (P.S) c condition if and only if every sequence {u j } in X such that lim j E (u j ) = c, and lim j E' (u j ) = 0 in X* has a convergent subsequence.

Lemma 3.2. Let (f1), (g1), (a1), (b1) hold. Then for any c ≠ 0, every bounded (P. S) c sequence for E, i.e., a bounded sequence {u n } X\{0} such that E (u n ) → c and E' (u n ) → 0, has a strongly convergent subsequence.

The proof of these two lemmas can be obtained easily from [25, 40], we omitted them here.

Theorem 3.1. Let (f1), (g1), (a1), (b1) and the following conditions hold true:

(a2) There are positive constants α1, M, and C such that a ^ ( t ) C t α 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq30_HTML.gif for tM.

(b2) There are positive constants β1 and C such that | b ^ ( t ) | C + C | t | β 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq31_HTML.gif for t R.

(H1) β1q1+ < α1p-, q2+ < α1p-.

Then the functional E is coercive and attains its infimum in X at some u0 X. Therefore, u0 is a solution of (P) if E is differentiable at u0.

Proof. For || u || large enough, by (f1), (g1), (a2), (b2) and (H1), we have that
J ( u ) = a ^ ( I 1 ( u ) ) = a ^ Ω 1 p ( x ) | u | p ( x ) + | u | p ( x ) d x a ^ ( C 1 | | u | | p - ) C 2 | | u | | α 1 p - , Ω F ( x , u ) d x C 3 | | u | | q 1 + , Φ ( u ) = b ^ ( I 2 ( u ) ) = b ^ Ω F ( x , u ) d x C 4 | | u | | β 1 q 1 + + C 4 ̃ , Ψ ( u ) = Ω G ( x , u ) d σ C 5 | | u | | q 2 + + C 5 ̃ , E ( u ) = J ( u ) - Φ ( u ) - Ψ ( u ) C 2 | | u | | α 1 p - - C 4 | | u | | β 1 q 1 + - C 5 | | u | | q 2 + - + C 6 ̃ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equq_HTML.gif

and hence E is coercive. Since E is sequentially weakly lower semi-continuous and X is reflexive, E attains its infimum in X at some u0 X. In this case E is differentiable at u0, then u0 is a solution of (P).

Theorem 3.2. Let (f1), (g1), (a1), (b1), (a2), (b2), (H1) and the following conditions hold true:

(a3) There is a positive constant α2 such that lim sup t 0 + a ^ ( t ) t α 2 < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq32_HTML.gif.

(b3) There is a positive constant β2 such that lim inf t 0 b ^ ( t ) | t | β 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq33_HTML.gif.

(f2) There exist an open subset Ω0 of Ω and r1 > 0 such that lim inf t 0 F ( x , t ) | t | r 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq34_HTML.gif uniformly for x Ω0.

(g2) There exists r2 > 0 such that lim inf t 0 G ( x , t ) | t | r 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq35_HTML.gif uniformly for x ∂Ω.

(H2) β2r1 < α2p-, r2 < α2p-.

Then (P) has at least one nontrivial solution which is a global minimizer of the energy functional E.

Proof. From Theorem 3.1 we know that E has a global minimizer u0. It is clear that a ^ ( 0 ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq36_HTML.gif b ^ ( 0 ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq37_HTML.gifF (x, 0) and consequently E (0) = 0. Take w C 0 ( Ω 0 ) \ { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq38_HTML.gif. Then, by (f2), (g2) (a3), (b3) and (H2), for sufficiently small λ > 0 we have that
E ( λ w ) = a ^ Ω λ p ( x ) p ( x ) | w | p ( x ) + | w | p ( x ) d x - b ^ Ω F ( x , λ w ) d x - Ω G ( x , λ w ) d σ C 1 Ω λ p ( x ) p ( x ) | w | p ( x ) + | w | p ( x ) d x α 2 - C 2 Ω 0 F ( x , λ w ) d x ) β 2 - C 3 Ω | λ w | r 2 d σ C 4 λ α 2 p - - C 5 λ β 2 r 1 - C 6 λ r 2 < 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equr_HTML.gif

Hence E (u0) < 0 and u0 ≠ 0.

By the genus theorem, similarly in the proof of Theorem 4.3 in [18], we have the following:

Theorem 3.3. Let the hypotheses of Theorem 3.2 hold, and let, in addition, f and g satisfy the following conditions:

(f3) f (x, - t) = - f (x, t) for x Ω and t R.

(g3) g (x, - t) = - g (x, t) for x ∂Ω and t R.

Then (P) has a sequence of solutions {u n } such that E(u n ) < 0.

Theorem 3.4. Let (f1), (g1), (a1), (b1), (a2), (b2), (a3), (b3), (H1), (H2) and the following conditions hold true:

(b+) b(t) ≥ 0 for t ≥ 0.

(f+) f(x, t) ≥ 0 for x Ω and t ≥ 0.

(g+) g(x, t) ≥ 0 for x ∂Ω and t ≥ 0.

(f2)+There exist an open subset Ω0 of Ω and r1 > 0 such that lim inf t 0 + F ( x , t ) t r 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq39_HTML.gif uniformly for x Ω0.

(g2)+ There exists r2 > 0 such that lim inf t 0 + G ( x , t ) t r 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq40_HTML.gif uniformly for x ∂Ω.

Then (P) has at least one nontrivial nonnegative solution with negative energy.

Proof. Define
f ̃ ( x , t ) = f ( x , t ) if  t 0 , f ( x , 0 ) if t < 0 , g ̃ ( x , t ) = g ( x , t ) if t 0 , g ( x , 0 ) if t < 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equs_HTML.gif
F ̃ ( x , t ) = 0 t f ̃ ( x , s ) d s , x Ω , t R , G ̃ ( x , t ) = 0 t g ̃ ( x , s ) d s , x Ω , t R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equt_HTML.gif
b ̃ ( t ) = b ( t ) if  t 0 , b ( 0 ) if  t < 0 , b ̃ ^ ( t ) = 0 t b ̃ ( s ) d s , t R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equu_HTML.gif
E ˜ ( u ) = a ^ ( Ω 1 p ( x ) ( | u | p ( x ) + | u | p ( x ) ) d x ) b ˜ ^ ( Ω F ˜ ( x , u ) d x ) Ω G ˜ ( x , u ) d σ , u X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equv_HTML.gif

Then, using truncation functions above, similarly in the proof of Theorem 3.4 in [25], we can prove that E ̃ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq41_HTML.gif has a nontrivial global minimizer u0 and u0 is a nontrivial nonnegative solution of (P).

4 The Mountain Pass theorem

In this section we will find the Mountain Pass type critical points of the energy functional E associated with problem (P).

Lemma 4.1. Let (f1), (g1), (a1), (b1) and the following conditions hold true:

( a 2 ) α 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq42_HTML.gif, M > 0, and C > 0 such that

a ^ ( t ) C t α 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq43_HTML.giffor all tM

with α1p-> 1.

(a4) λ > 0, M > 0 such that

λ a ^ ( t ) a ( t ) t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq44_HTML.giffor all tM

(b4) θ > 0, M > 0 such that:

0 θ b ^ ( t ) b ( t ) t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq45_HTML.gif, for all tM.

(f4) μ > 0, M > 0 such that:

0 ≤ μF(x, t) ≤ f(x, t)t, for |t| ≥ M and x Ω.

(g4) κ > θμ > 0, M > 0 such that:

0 ≤ κG(x, t) ≤ g(x, t)t, |t| ≥ M and x Ω.

(H3) λp+< θμ.

Then E satisfies condition (P.S)c for any c ≠ 0.

Proof. By (a4), for ||u|| large enough,
λ p + J ( u ) = λ p + a ^ ( Ω 1 p ( x ) ( | u | p ( x ) + | u | p ( x ) ) d x ) p + a ( Ω 1 p ( x ) ( | u | p ( x ) + | u | p ( x ) ) d x ) Ω 1 p ( x ) ( | u | p ( x ) + | u | p ( x ) ) d x a ( Ω 1 p ( x ) ( | u | p ( x ) + | u | p ( x ) ) d x ) Ω ( | u | p ( x ) + | u | p ( x ) ) d x = J ( u ) u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equw_HTML.gif
From (f4) and (g4) we can see that there exists C1> 0 and C2> 0 such that
- C 1 μ Ω F ( x , u ) d x Ω f ( x , u ) u d x + C 1 , u X , - C 2 κ Ω G ( x , u ) d σ Ω g ( x , u ) u d σ + C 2 , u X , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equx_HTML.gif
and thus, given any ε (0, μ), there exists M ε M > 0 and M ε M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq46_HTML.gif such that
( μ - ε ) Ω F ( x , u ) d x Ω f ( x , u ) u d x , if Ω F ( x , u ) d x M ε , θ ( μ - ε ) Ω G ( x , u ) d σ Ω g ( x , u ) u d σ , if Ω G ( x , u ) d σ M ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equy_HTML.gif
We may assume M ε > c 1 μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq47_HTML.gif and M ε > c 2 θ μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq48_HTML.gif. Note that in this case the inequalities Ω F ( x , u ) d x M ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq49_HTML.gif and Ω G ( x , u ) d σ M ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq50_HTML.gif are equivalent to | Ω F ( x , u ) d x | M ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq51_HTML.gifand | Ω G ( x , u ) d σ | M ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq52_HTML.gif, because Ω F ( x , u ) d x - C 1 μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq53_HTML.gif and Ω G ( x , u ) d σ - c 2 θ μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq54_HTML.gif for all u X. We claim that there exist C ε > 0 and C ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq55_HTML.gif such that
Φ ( u ) u - θ ( μ - ε ) Φ ( u ) - C ε  for  u X , Ψ ( u ) u - θ ( μ - ε ) Ψ ( u ) - C ε  for  u X . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equz_HTML.gif
Indeed, when | Ω F ( x , u ) d x | M ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq56_HTML.gif and | Ω G ( x , u ) d σ | M ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq57_HTML.gif, the validity is obvious. When | Ω F ( x , u ) d x | M ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq58_HTML.gif and | Ω G ( x , u ) d σ | M ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq59_HTML.gif, i.e., Ω F ( x , u ) d x M ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq60_HTML.gif and Ω G ( x , u ) d σ M ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq61_HTML.gif, we have that
θ ( μ - ε ) Φ ( u ) = θ ( μ - ε ) b ^ Ω F ( x , u ) d x ( μ - ε ) b Ω F ( x , u ) d x Ω F ( x , u ) d x b Ω F ( x , u ) d x Ω f ( x , u ) u d x = Φ ( u ) u , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equaa_HTML.gif
and
θ ( μ - ε ) Ψ ( u ) = θ ( μ - ε ) Ω G ( x , u ) d σ Ω g ( x , u ) u d σ = Ψ ( u ) u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equab_HTML.gif
Now let {u n } X\{0}, E(u n ) → c ≠ 0 and E'(u n ) → 0. By (H3), there exists ε > 0 small enough such that λp+< θ(μ - ε). Then, since {u n } is a (P.S) c sequence, for sufficiently large n, we have
θ ( μ - ε ) c + 1 + u n θ ( μ - ε ) E ( u n ) - E ( u n ) u n ( θ ( μ - ε ) - λ p + ) J ( u n ) + ( λ p + J ( u n ) - J ( u n ) u n ) + ( Φ ( u n ) u n - θ ( μ - ε ) Φ ( u n ) ) + ( Ψ ( u n ) u n - θ ( μ - ε ) Ψ ( u n ) ) C 3 u n α 1 p - - C 4 - C ε - C ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_Equac_HTML.gif

Since α1p-> 1, we have that {||u n ||} is bounded. By Lemma 3.2, E satisfies condition (P.S) c for c ≠ 0.

Theorem 4.1. Under the hypotheses of Lemma 4.1, and let the following conditions hold:

(a5) There is a positive constant α3 such that lim sup t 0 + a ^ ( t ) t α 3 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq62_HTML.gif.

(b5) There is a positive constant β3 such that lim inf t 0 b ^ ( t ) | t | β 3 < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq63_HTML.gif.

(f5) There exists r 1 C 0 ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq64_HTML.gif such that 1 < r1(x) < p*(x) for x Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq65_HTML.gif and lim inf t 0 | F ( x , t ) | | t | r 1 ( x ) < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq66_HTML.gif uniformly for x Ω.

(g5) There exists such r 2 C 0 ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq67_HTML.gif such that 1 < r2(x) < p*(x) for x Ω and lim inf t 0 | G ( x , t ) | | t | r 2 ( x ) < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq68_HTML.gif uniformly for x Ω.

(H4) α3p+< β3r1-, α3p+< r2-, λp+< θμ.

Then (P) has a nontrivial solution with positive energy.

Proof. Let us prove this conclusion by the Mountain Pass lemma. E satisfies condition (P.S) c for c ≠ 0 has been proved in Lemma 4.1.

For ||u|| small enough, from (a5) we can obtain easily that J ( u ) C 1 u α 3 p + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq69_HTML.gif, from (b5), (f1) and (f5) we have | Φ ( u ) | C 2 u β 3 r 1 - https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq70_HTML.gif, and in the similar way from(g1) and (g5) we have | Ψ ( u ) | C 2 u r 2 - https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq71_HTML.gif. Thus by (H4), we conclude that there exist positive constants ρ and δ such that E(u) ≥ for ||u|| = ρ.

Let w X\{0} be given. From (a4) for sufficiently large t > 0 we have a ^ ( t ) C 1 t λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq72_HTML.gif, which follows that J ( s w ) d 1 s λ p + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq73_HTML.gif for s large enough, where d1 is a positive constant depending on w. From (f4) and (f1) for |t| large enough we have Ω F ( x , s w ) d x d 2 s μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq74_HTML.gif for s large enough, where d2 is a positive constant depending on w. From (b4) for t large enough we have Φ ( s w ) = b ^ ( Ω F ( x , s w ) d x ) d 3 s θ μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq75_HTML.gif for s large enough, where d3 is a positive constant depending on w. From (g4) and (g1) for |t| large enough we have Ψ ( s w ) = Ω G ( x , s w ) d σ d 4 s θ μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq76_HTML.gif. Hence for any w X\{0} and s large enough, E ( s w ) d 1 s λ p + - d 3 s θ μ - d 4 s θ μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-1/MediaObjects/13661_2011_Article_105_IEq77_HTML.gif, thus by (H3), We conclude that E(sw) → -∞ as s → +∞.

So by the Mountain Pass lemma this theorem is proved.

By the symmetric Mountain Pass lemma, similarly in the proof of Theorem 4.8 in [40], we have the following:

Theorem 4.2. Under the hypotheses of Theorem 4.1, if, in addition, (f3) and (g3) are satisfied, then (P) has a sequence of solutions {±u n } such that Eu n ) → +∞ as n → ∞.

Declarations

Acknowledgements

The authors thank the two referees for their careful reading and helpful comments of the study. Research supported by the National Natural Science Foundation of China (10971088), (10971087).

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Lanzhou University

References

  1. Kirchhoff G: Mechanik. Teubner, Leipzig; 1883.
  2. Lions JL: On some questions in boundary value problems of mathematical physics. In Proceedings of International Symposium on Continuum Mechanics and Partial Differential Equations Math Stud North-Holland Edited by: Rio de Janeiro 1977, in: de la Penha, Medeiros. 1978, 30: 284–346.View Article
  3. Arosio A, Panizzi S: On the well-posedness of the Kirchhoff string.Trans Am Math Soc 1996, 348: 305–330. 10.1090/S0002-9947-96-01532-2MathSciNetView Article
  4. Cavalcanti MM, Domingos Cavalcanti VN, Soriano JA: Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation.Adv Diff Equ 2001, 6: 701–730.MathSciNet
  5. Chipot M, Lovat B: Some remarks on non local elliptic and parabolic problems.Nonlinear Anal 1997, 30: 4619–4627. 10.1016/S0362-546X(97)00169-7MathSciNetView Article
  6. D'Ancona P, Spagnolo S: Global solvability for the degenerate Kirchhoff equation with real analytic date.Invent Math 1992, 108: 447–462.MathSciNetView Article
  7. Chipot M, Rodrigues JF: On a class of nonlocal nonlinear elliptic problems.RAIRO Modélisation Math Anal Numbér 1992, 26: 447–467.MathSciNet
  8. Dreher M: The Kirchhoff equation for the p-Laplacian.Rend Semin Mat Univ Politec Torino 2006, 64: 217–238.MathSciNet
  9. Dreher M: The wave equation for the p-Laplacian.Hokkaido Math J 2007, 36: 21–52.MathSciNetView Article
  10. Autuori G, Pucci P, Salvatori MC: Asymptotic stability for anistropic Kirchhoff systems.J Math Anal Appl 2009, 352: 149–165. 10.1016/j.jmaa.2008.04.066MathSciNetView Article
  11. Perera K, Zhang ZT: Nontrivial solutions of Kirchhoff-type problems via the Yang index.J Diff Equ 2006, 221: 246–255. 10.1016/j.jde.2005.03.006MathSciNetView Article
  12. Alves CO, Corrêa FJSA, Ma TF: Positive solutions for a quasilinear elliptic equation of Kirchhoff type.Comput Math Appl 2005, 49: 85–93. 10.1016/j.camwa.2005.01.008MathSciNetView Article
  13. Corrêa FJSA, Figueiredo GM: On an elliptic equation of p-Kirchhoff type via variational methods.Bull Aust Math Soc 2006, 74: 263–277. 10.1017/S000497270003570XView Article
  14. Corrêa FJSA, Figueiredo GM: On a p-Kirchhoff equation via Krasnosel-skii's genus.Appl Math Lett 2009, 22: 819–822. 10.1016/j.aml.2008.06.042MathSciNetView Article
  15. Corrêa FJSA, Menezes SDB, Ferreira J: On a class of problems involving a nonlocal operator.Appl Math Comput 2004, 147: 475–489. 10.1016/S0096-3003(02)00740-3MathSciNetView Article
  16. He XM, Zou WM: Infinitly many positive solutions for Kirchhoff-type problems.Nonlinear Anal 2009, 70: 1407–1414. 10.1016/j.na.2008.02.021MathSciNetView Article
  17. Fan XL: On the sub-supersolution method forp(x)-Laplacian equations.J Math Anal Appl 2007, 330: 665–682. 10.1016/j.jmaa.2006.07.093MathSciNetView Article
  18. Fan XL, Han XY: Existence and multiplicity of solutions forp(x)-Laplacian equations in R n .Nonlinear Anal 2004, 59: 173–188.MathSciNet
  19. Fan XL, Shen JS, Zhao D: Sobolev embedding theorems for spaceWk, p(x)(Ω).J Math Anal Appl 2001, 262: 749–760. 10.1006/jmaa.2001.7618MathSciNetView Article
  20. Fan XL, Zhang QH: Existence of solutions forp(x)-Laplacian Dirichlet problems.Nonlinear Anal 2003, 52: 1843–1852. 10.1016/S0362-546X(02)00150-5MathSciNetView Article
  21. Fan XL, Zhang QH, Zhao D: Eigenvalues ofp(x)-Laplacian Dirichlet problem.J Math Anal Appl 2005, 302: 306–317. 10.1016/j.jmaa.2003.11.020MathSciNetView Article
  22. Fan XL, Zhao D: On the spacesLp(x)(Ω) andWk, p(x)(Ω).J Math Anal Appl 2001, 263: 424–446. 10.1006/jmaa.2000.7617MathSciNetView Article
  23. Fan XL, Zhao YZ, Zhang QH: A strong maximum principle forp(x)-Laplacian equations.Chinese Ann Math Ser A 2003, 24: 495–500. (in Chinese); Chinese J Contemp Math 24: 277–282 (2003)MathSciNet
  24. Dai GW, Hao RF: Existence of solutions for ap(x)-Kirchhoff-type equation.J Math Anal Appl 2009, 359: 275–284. 10.1016/j.jmaa.2009.05.031MathSciNetView Article
  25. Fan XL: On nonlocalp(x)-Laplacian Dirichlet problems.Nonlinear Anal 2010, 72: 3314–3323. 10.1016/j.na.2009.12.012MathSciNetView Article
  26. Chipot M, Shafrir I, Fila M: On the solutions to some elliptic equations with nonlinear boundary conditions.Adv Diff Eq 1996, 1: 91–110.MathSciNet
  27. Hu B: Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition.Diff Integral Equ 1994, 7(2):301–313.
  28. dal Maso Gianni, Ebobisse Francois, Ponsiglione Marcello: A stability result for nonlinear Neumann problems under boundary variations.J Math Pures Appl 2003, 82: 503–532. 10.1016/S0021-7824(03)00014-XMathSciNetView Article
  29. Garcia-Azorero J, Peral I, Rossi JD: A convex-concave problem with a nonlinear boundary condition.J Diff Equ 2004, 198: 91–128. 10.1016/S0022-0396(03)00068-8MathSciNetView Article
  30. Song XC, Wang WH, Zhao PH: Positive solutions of elliptic equations with nonlinear boundary conditions.Nonlinear Anal 2009, 70: 328–334. 10.1016/j.na.2007.12.003MathSciNetView Article
  31. Bonder JF, Pinasco JP, Rossi JD: Existence results for Hamiltonian elliptic systems with nonlinear boundary conditions.Electron J Diff Equ 1999, 40: 1–15.
  32. Bonder JF, Rossi JD: Existence for an elliptic system with nonlinear boundary conditions via fixed point methods.Adv Diff Equ 2001, 6: 1–20.
  33. Bonder JF, Rossi JD: Existence results for thep-Laplacian with nonlinear boundary conditions.J Math Anal Appl 2001, 263: 195–223. 10.1006/jmaa.2001.7609MathSciNetView Article
  34. Cîrstea Florica-CorinaŞT, Rădulescu VicenţiuD: Existence and nonexistence results for a quasilinear problem with nonlinear boundary condition.J Math Anal Appl 2000, 244: 169–183. 10.1006/jmaa.1999.6699MathSciNetView Article
  35. Afrouzi GA, Alizadeh M: A quasilinearization method forp-Laplacian equations with a nonlinear boundary condition.Nonlinear Anal 2009, 71: 2829–2833. 10.1016/j.na.2009.01.134MathSciNetView Article
  36. Martinez S, Rossi JD: On the Fučik spectrum and a resonance problem for thep-Laplacian with a nonlinear boundary condition.Nonlinear Anal 2004, 59: 813–848.MathSciNet
  37. Afrouzi GA, Rasouli SH: A variational approach to a quasilinear elliptic problem involving thep-Laplacian and nonlinear boundary condition.Nonlinear Anal 2009, 71: 2447–2455. 10.1016/j.na.2009.01.090MathSciNetView Article
  38. Deng SG, Wang Q: Nonexistence, existence and multiplicity of positive solutions to thep(x)-Laplacian nonlinear Neumann boundary value problem.Nonlinear Anal 2010, 73: 2170–2183. 10.1016/j.na.2010.05.043MathSciNetView Article
  39. Deng SG: A local mountain pass theorem and applications to a double perturbedp(x)-Laplacian equations.Appl Math Comput 2009, 211: 234–241. 10.1016/j.amc.2009.01.042MathSciNetView Article
  40. Yao JH: Solutions for Neumann boundary value problems involvingp(x)-Laplace operators.Nonlinear Anal 2008, 68: 1271–1283. 10.1016/j.na.2006.12.020MathSciNetView Article
  41. Edmunds DE, Rákosník J: Density of smooth functions inWk, p(x)(Ω).Proc R Soc A 1992, 437: 229–236. 10.1098/rspa.1992.0059View Article
  42. Edmunds DE, Rákosník J: Sobolev embedding with variable exponent.Studia Math 2000, 143: 267–293.MathSciNet

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