SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

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

Erlin Guo* and Peihao Zhao

Author Affiliations

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P. R. China

For all author emails, please log on.

Boundary Value Problems 2012, 2012:1  doi:10.1186/1687-2770-2012-1


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/1


Received:29 August 2011
Accepted:4 January 2012
Published:4 January 2012

© 2012 Guo and Zhao; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M1">View MathML</a>

where Ω is a smooth bounded domain and ν is the outward normal vector on the boundary ∂Ω, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M2">View MathML</a>. 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M3">View MathML</a>

where Ω is a smooth bounded domain in RN, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M4">View MathML</a> with 1 < p- := infΩ p(x) ≤ p(x) ≤ p+ := supΩ p(x) < N, a(t) is a continuous real-valued function, f : Ω × R R, g : ∂Ω × R R satisfy the Caratheodory condition, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M5">View MathML</a>. 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M6">View MathML</a>

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. [3-6] 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M7">View MathML</a>

The study of the stationary version of Kirchhoff type problems has received considerable attention in recent years; see e.g. [5,11-16].

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 [17-23] 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 [26-30], for elliptic systems with nonlinear boundary conditions see [31,32], for the p-Laplacian with nonlinear boundary conditions of different type see [33-37], and for the p(x)-Laplacian with nonlinear boundary conditions see [38-40]. 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 RN, denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M8">View MathML</a>

we can introduce the norm on Lp(x) (Ω) by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M9">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M10">View MathML</a>

and it can be equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M11">View MathML</a>

where |∇u|p(x) = ||∇u||p(x); and we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M12">View MathML</a> the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M13">View MathML</a> in W1, p(x)(Ω), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M15">View MathML</a>, when p(x) < N, and p* = p* = ∞, when p(x) > N.

Proposition 2.1 [22,41]. (1) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M16">View MathML</a>, 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M17">View MathML</a>

(2) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M18">View MathML</a>, p1 (x) ≤ p2 (x), for any x ∈ Ω, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M19">View MathML</a>, and the imbedding is continuous.

Proposition 2.2 [22]. If f : Ω × R R is a Caratheodory function and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M20">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M22">View MathML</a>, d(x) ≥ 0 and e ≥ 0 is a constant, then the superposition operator from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M23">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M24">View MathML</a> defined by (Nf (u)) (x) = f (x, u (x)) is a continuous and bounded operator.

Proposition 2.3 [22]. If we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M25">View MathML</a>

then for u, un Lp(x) (Ω)

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

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M26">View MathML</a>

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M27">View MathML</a>

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

(1) limk → ∞ |uk - u|p(x) = 0;

(2) limk → ∞ ρ |uk - u| = 0;

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

Proposition 2.5 [22]. (1) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M28">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M29">View MathML</a> and W1,p(x)(Ω) are separable reflexive Banach spaces;

(2) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M30">View MathML</a> and q (x) < p* (x) for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M31">View MathML</a>, then the imbedding from W1, p(x)(Ω) to Lq(x) (Ω) is compact and continuous;

(3) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M30">View MathML</a> and q (x) < p* (x) for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M31">View MathML</a>, 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M32">View MathML</a>

So, |∇u|p(x) is a norm equivalent to the norm || u || in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M33">View MathML</a>.

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 : R R is continuous.

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

(fl) f : Ω × R R satisfies the Caratheodory condition and there exist two constants C1 ≥ 0, C2 ≥ 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M34">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M35">View MathML</a> and q1 (x) < p* (x), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M36">View MathML</a>.

(g1) g : ∂Ω × R R satisfies the Caratheodory condition and there exist two constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M37">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M38">View MathML</a>

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 onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M39">View MathML</a>,

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M40">View MathML</a>.

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

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M41">View MathML</a>.

(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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M42">View MathML</a>

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 : X R is sequentially weakly lower semi-continuous, Φ, Ψ: X R 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, un u in X implies Φ' (un) → Φ' (u) in X*. For any open set D ⊂ X\{0} with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M43">View MathML</a>, The mappings J' and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M44">View MathML</a> are bounded, and are of type (S+), namely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M45">View MathML</a>

Definition 3.1. Let c ∈ R, a C1-functional E : X R satisfies (P.S)c condition if and only if every sequence {uj} in X such that limj E (uj) = c, and limj E' (uj) = 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 {un} ⊂ X\{0} such that E (un) → c and E' (un) → 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 onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M46">View MathML</a> for t M.

(b2) There are positive constants β1 and C such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M47">View MathML</a> for t R.

(H1) β1 q1+ < α1 p-, 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M48">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M49">View MathML</a>.

(b3) There is a positive constant β2 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M50">View MathML</a>.

(f2) There exist an open subset Ω0 of Ω and r1 > 0 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M51">View MathML</a> uniformly for x ∈ Ω0.

(g2) There exists r2 > 0 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M52">View MathML</a> uniformly for x ∈ ∂Ω.

(H2) β2r1 < α2 p-, r2 < α2 p-.

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 onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M53">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M54">View MathML</a>F (x, 0) and consequently E (0) = 0. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M55">View MathML</a>. Then, by (f2), (g2) (a3), (b3) and (H2), for sufficiently small λ > 0 we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M56">View MathML</a>

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 {un} such that E(un) < 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M57">View MathML</a> uniformly for x ∈ Ω0.

(g2)+ There exists r2 > 0 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M58">View MathML</a> uniformly for x ∈ ∂Ω.

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

Proof. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M59">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M60">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M61">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M62">View MathML</a>

Then, using truncation functions above, similarly in the proof of Theorem 3.4 in [25], we can prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M63">View MathML</a> 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 onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M64">View MathML</a>, M > 0, and C > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M65">View MathML</a>for all t M

with α1p- > 1.

(a4) ∃ λ > 0, M > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M66">View MathML</a>for all t M

(b4) ∃θ > 0, M > 0 such that:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M67">View MathML</a>, for all t M.

(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,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M68">View MathML</a>

From (f4) and (g4) we can see that there exists C1 > 0 and C2 > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M69">View MathML</a>

and thus, given any ε ∈ (0, μ), there exists Mε M > 0 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M70">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M71">View MathML</a>

We may assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M72">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M73">View MathML</a>. Note that in this case the inequalities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M74">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M75">View MathML</a> are equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M76">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M77">View MathML</a>, because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M78">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M79">View MathML</a> for all u X. We claim that there exist Cε > 0 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M80">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M81">View MathML</a>

Indeed, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M82">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M83">View MathML</a>, the validity is obvious. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M84">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M85">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M87">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M88">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M89">View MathML</a>

Now let {un} ⊂ X\{0}, E(un) → c ≠ 0 and E'(un) → 0. By (H3), there exists ε > 0 small enough such that λp+ < θ(μ - ε). Then, since {un} is a (P.S)c sequence, for sufficiently large n, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M90">View MathML</a>

Since α1p- > 1, we have that {||un||} 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M91">View MathML</a>.

(b5) There is a positive constant β3 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M92">View MathML</a>.

(f5) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M93">View MathML</a> such that 1 < r1(x) < p*(x) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M94">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M95">View MathML</a> uniformly for x ∈ Ω.

(g5) There exists such <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M96">View MathML</a> such that 1 < r2(x) < p*(x) for x Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M97">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M98">View MathML</a>, from (b5), (f1) and (f5) we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M99">View MathML</a>, and in the similar way from(g1) and (g5) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M100">View MathML</a>. 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 onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M101">View MathML</a>, which follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M102">View MathML</a> for s large enough, where d1 is a positive constant depending on w. From (f4) and (f1) for |t| large enough we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M103">View MathML</a> for s large enough, where d2 is a positive constant depending on w. From (b4) for t large enough we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M104">View MathML</a> for s large enough, where d3 is a positive constant depending on w. From (g4) and (g1) for |t| large enough we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M105">View MathML</a>. Hence for any w X\{0} and s large enough, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/1/mathml/M106">View MathML</a>, 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 {±un} such that Eun) → +∞ as n → ∞.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

EG and PZ contributed to each part of this work equally. All the authors read and approved the final manuscript.

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).

References

  1. Kirchhoff, G: Mechanik. Teubner, Leipzig (1883)

  2. Lions, JL: On some questions in boundary value problems of mathematical physics. vol. 30, In: Rio de Janeiro 1977, in: de la Penha, Medeiros (eds.) Proceedings of International Symposium on Continuum Mechanics and Partial Differential Equations Math Stud North-Holland. 30, 284–346 (1978)

  3. Arosio, A, Panizzi, S: On the well-posedness of the Kirchhoff string. Trans Am Math Soc. 348, 305–330 (1996). Publisher Full Text OpenURL

  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. 6, 701–730 (2001)

  5. Chipot, M, Lovat, B: Some remarks on non local elliptic and parabolic problems. Nonlinear Anal. 30, 4619–4627 (1997). Publisher Full Text OpenURL

  6. D'Ancona, P, Spagnolo, S: Global solvability for the degenerate Kirchhoff equation with real analytic date. Invent Math. 108, 447–462 (1992)

  7. Chipot, M, Rodrigues, JF: On a class of nonlocal nonlinear elliptic problems. RAIRO Modélisation Math Anal Numbér. 26, 447–467 (1992). PubMed Abstract OpenURL

  8. Dreher, M: The Kirchhoff equation for the p-Laplacian. Rend Semin Mat Univ Politec Torino. 64, 217–238 (2006)

  9. Dreher, M: The wave equation for the p-Laplacian. Hokkaido Math J. 36, 21–52 (2007)

  10. Autuori, G, Pucci, P, Salvatori, MC: Asymptotic stability for anistropic Kirchhoff systems. J Math Anal Appl. 352, 149–165 (2009). Publisher Full Text OpenURL

  11. Perera, K, Zhang, ZT: Nontrivial solutions of Kirchhoff-type problems via the Yang index. J Diff Equ. 221, 246–255 (2006). Publisher Full Text OpenURL

  12. Alves, CO, Corrêa, FJSA, Ma, TF: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput Math Appl. 49, 85–93 (2005). Publisher Full Text OpenURL

  13. Corrêa, FJSA, Figueiredo, GM: On an elliptic equation of p-Kirchhoff type via variational methods. Bull Aust Math Soc. 74, 263–277 (2006). Publisher Full Text OpenURL

  14. Corrêa, FJSA, Figueiredo, GM: On a p-Kirchhoff equation via Krasnosel-skii's genus. Appl Math Lett. 22, 819–822 (2009). Publisher Full Text OpenURL

  15. Corrêa, FJSA, Menezes, SDB, Ferreira, J: On a class of problems involving a nonlocal operator. Appl Math Comput. 147, 475–489 (2004). Publisher Full Text OpenURL

  16. He, XM, Zou, WM: Infinitly many positive solutions for Kirchhoff-type problems. Nonlinear Anal. 70, 1407–1414 (2009). Publisher Full Text OpenURL

  17. Fan, XL: On the sub-supersolution method for p(x)-Laplacian equations. J Math Anal Appl. 330, 665–682 (2007). Publisher Full Text OpenURL

  18. Fan, XL, Han, XY: Existence and multiplicity of solutions for p(x)-Laplacian equations in Rn. Nonlinear Anal. 59, 173–188 (2004)

  19. Fan, XL, Shen, JS, Zhao, D: Sobolev embedding theorems for space Wk,p(x)(Ω). J Math Anal Appl. 262, 749–760 (2001). Publisher Full Text OpenURL

  20. Fan, XL, Zhang, QH: Existence of solutions for p(x)-Laplacian Dirichlet problems. Nonlinear Anal. 52, 1843–1852 (2003). Publisher Full Text OpenURL

  21. Fan, XL, Zhang, QH, Zhao, D: Eigenvalues of p(x)-Laplacian Dirichlet problem. J Math Anal Appl. 302, 306–317 (2005). Publisher Full Text OpenURL

  22. Fan, XL, Zhao, D: On the spaces Lp(x)(Ω) and Wk,p(x)(Ω). J Math Anal Appl. 263, 424–446 (2001). Publisher Full Text OpenURL

  23. Fan, XL, Zhao, YZ, Zhang, QH: A strong maximum principle for p(x)-Laplacian equations. Chinese Ann Math Ser A. 24, 495–500 (in Chinese); Chinese J Contemp Math 24: 277-282 (2003) (2003)

  24. Dai, GW, Hao, RF: Existence of solutions for a p(x)-Kirchhoff-type equation. J Math Anal Appl. 359, 275–284 (2009). Publisher Full Text OpenURL

  25. Fan, XL: On nonlocal p(x)-Laplacian Dirichlet problems. Nonlinear Anal. 72, 3314–3323 (2010). Publisher Full Text OpenURL

  26. Chipot, M, Shafrir, I, Fila, M: On the solutions to some elliptic equations with nonlinear boundary conditions. Adv Diff Eq. 1, 91–110 (1996)

  27. Hu, B: Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition. Diff Integral Equ. 7(2), 301–313 (1994)

  28. dal Maso, Gianni, Ebobisse, Francois, Ponsiglione, Marcello: A stability result for nonlinear Neumann problems under boundary variations. J Math Pures Appl. 82, 503–532 (2003). Publisher Full Text OpenURL

  29. Garcia-Azorero, J, Peral, I, Rossi, JD: A convex-concave problem with a nonlinear boundary condition. J Diff Equ. 198, 91–128 (2004). Publisher Full Text OpenURL

  30. Song, XC, Wang, WH, Zhao, PH: Positive solutions of elliptic equations with nonlinear boundary conditions. Nonlinear Anal. 70, 328–334 (2009). Publisher Full Text OpenURL

  31. Bonder, JF, Pinasco, JP, Rossi, JD: Existence results for Hamiltonian elliptic systems with nonlinear boundary conditions. Electron J Diff Equ. 40, 1–15 (1999)

  32. Bonder, JF, Rossi, JD: Existence for an elliptic system with nonlinear boundary conditions via fixed point methods. Adv Diff Equ. 6, 1–20 (2001)

  33. Bonder, JF, Rossi, JD: Existence results for the p-Laplacian with nonlinear boundary conditions. J Math Anal Appl. 263, 195–223 (2001). Publisher Full Text OpenURL

  34. Cîrstea ŞT., Florica-Corina, Rădulescu D., Vicenţiu: Existence and nonexistence results for a quasilinear problem with nonlinear boundary condition. J Math Anal Appl. 244, 169–183 (2000). Publisher Full Text OpenURL

  35. Afrouzi, GA, Alizadeh, M: A quasilinearization method for p-Laplacian equations with a nonlinear boundary condition. Nonlinear Anal. 71, 2829–2833 (2009). Publisher Full Text OpenURL

  36. Martinez, S, Rossi, JD: On the Fučik spectrum and a resonance problem for the p-Laplacian with a nonlinear boundary condition. Nonlinear Anal. 59, 813–848 (2004)

  37. Afrouzi, GA, Rasouli, SH: A variational approach to a quasilinear elliptic problem involving the p-Laplacian and nonlinear boundary condition. Nonlinear Anal. 71, 2447–2455 (2009). Publisher Full Text OpenURL

  38. Deng, SG, Wang, Q: Nonexistence, existence and multiplicity of positive solutions to the p(x)-Laplacian nonlinear Neumann boundary value problem. Nonlinear Anal. 73, 2170–2183 (2010). Publisher Full Text OpenURL

  39. Deng, SG: A local mountain pass theorem and applications to a double perturbed p(x)-Laplacian equations. Appl Math Comput. 211, 234–241 (2009). Publisher Full Text OpenURL

  40. Yao, JH: Solutions for Neumann boundary value problems involving p(x)-Laplace operators. Nonlinear Anal. 68, 1271–1283 (2008). Publisher Full Text OpenURL

  41. Edmunds, DE, Rákosník, J: Density of smooth functions in Wk,p(x)(Ω). Proc R Soc A. 437, 229–236 (1992). Publisher Full Text OpenURL

  42. Edmunds, DE, Rákosník, J: Sobolev embedding with variable exponent. Studia Math. 143, 267–293 (2000)