SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Three solutions for a class of quasilinear elliptic systems involving the p(x)-Laplace operator

Honghui Yin12* and Zuodong Yang13

Author Affiliations

1 Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Jiangsu Nanjing 210046, China

2 School of Mathematical Sciences, Huaiyin Normal University, Jiangsu Huaian 223001, China

3 College of Zhongbei, Nanjing Normal University, Jiangsu Nanjing 210046, China

For all author emails, please log on.

Boundary Value Problems 2012, 2012:30  doi:10.1186/1687-2770-2012-30

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/30


Received:6 October 2011
Accepted:7 March 2012
Published:7 March 2012

© 2012 Yin and Yang; 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

The existence of at least three weak solutions is established for a class of quasilinear elliptic systems involving the p(x)-Laplace operator with Neumann boundary condition. The technical approach is mainly based on a three critical points theorem due to Ricceri.

MSC: 35D05; 35J60; 58E05.

Keywords:
p(x)-Laplacian; Sobolev space; three critical points theorem

1 Introduction

In this article, we consider the problem of the type

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M1">View MathML</a>

(1)

where Ω ⊂ RN(N ≥ 2) is a bounded domain with boundary of class C1. ν is the outer unit normal to ∂Ω, λ, μ ≥ 0 are real numbers. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M2">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M3">View MathML</a>, N < q- q+, F : Ω × R × R R is a function such that F(·, s, t) is measurable in Ω for all (s, t) ∈ R × R and F(x, ·, ·) is C1 in R × R for a.e. x ∈ Ω, Fs denotes the partial derivative of F with respect to s. We assume G(x,s,t) and ep(x),eq(x) satisfy the following conditions:

(G) G : Ω × R × R R is a Carathéodory function, sup{|s|≤θ,|t|≤ϑ} |G(·,s,t)| ∈ L1(Ω) for all θ, ϑ > 0;

(E) ep(x),eq(x) ∈ L(Ω) and ess infep(x), ess infeq(x) > 0, we denote ∥ep1 = ∫ep(x)dx and ∥eq1 = ∫eq(x)dx.

It is well known that the operator -Δp(x) = -div(|∇u|p(x)-2u) is called p(x)-Laplacian and the corresponding problem is called a variable exponent elliptic systems. The study of differential equations and variational problems with nonstandard p(x)-growth conditions has been attracting attention of many authors in the last two decades. It arises from nonlinear elasticity theory, electro-rheological fluids, etc. see [1,2], many results have been obtained on this kind of problems, for example [3-9]. For the special case, p(x) ≡ p(a constant), (1.1) becomes the well known p-Laplacian problem. There have been many papers on this class of problems, see [10-19] and the reference therein.

Recently, many papers have appeared in which the technical approach adopted is based on the three critical points theorem obtained by Ricceri [16]. We cite papers [20-23], where the authors established the existence of at least three weak solutions to the problems with Dirichlet or Neumann boundary value conditions. Li and Tang in [24] obtained the existence of at least three weak solutions to problem (1) when p(x) ≡ p with Dirichlet boundary value conditions. El Manouni and Kbiri Alaoui [25] obtained the existence of at least three solutions of system (1) when p(x) ≡ p in Ω by the three critical points theorem obtained by Ricceri [26].

The main purpose of the present paper is to prove the existence of at least three solutions of problem (1). We study problem (1) by using the three critical points theorem by Ricceri [26] too. On the basis of [27], we state an equivalent formulation of the three critical points theorem in [26] as follows.

Theorem 1. Let X be a reflexive real Banach space, Φ : X R a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous C1 functional, bounded on each bounded subset of X, whose Gâteaux derivative admits a continuous inverse on X*; Ψ : X R a C1 functional with compact Gâteaux derivative. Assume that

(i) limu∥→∞(Φ(u) + λ Ψ(u)) = ∞ for all λ > 0; and there are r R and u0, u1 X such that:

(ii) Φ(u0) < r < Φ(u1);

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M4">View MathML</a>.

Then there exists a non-empty open set Λ ⊆ [0, ∞) and a positive real number ρ with the following property: for each λ ∈ Λ and every C1 functional J : X R with compact Gâteaux derivative, there exists σ > 0 such that for each μ ∈ [0, σ], the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M5">View MathML</a>

(2)

has at least three solutions in X whose norms are less than ρ.

The paper is organized as follows. In section 2, we recall some facts that will be needed in the paper. In section 3, we establish our main result.

2 Notations and preliminaries

In order to deal with p(x)-Laplacian problem, we need some theories on spaces Lp(x)(Ω), W1,p(x)(Ω) and properties of p(x)-Laplacian which we will use later (see [1,5,28,29]).

We denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M6">View MathML</a>

We can introduce a norm on Lp(x)(Ω) by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M7">View MathML</a>

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

The space W1,p(x)(Ω) is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M8">View MathML</a>

and it can be equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M9">View MathML</a>

and we call it variable exponent Sobolev space. From [5], we know that spaces Lp(x)(Ω) and W1,p(x)(Ω) are separable, reflexive and uniform convex Banach spaces.

When ep(x) satisfy (E), we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M10">View MathML</a>

with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M11">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M12">View MathML</a> is a Banach space. For any u W1,p(x)(Ω), define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M13">View MathML</a>

Then it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M14">View MathML</a> is a norm on W1,p(x)(Ω) equivalent to ∥up(x). In the following, we will use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M15">View MathML</a> to instead of ∥ · ∥p(x) on W1,p(x)(Ω). Similarly, we use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M15">View MathML</a> to instead of ∥ · ∥q(x) on W1,q(x)(Ω).

Proposition 1. (see [1,5]) The conjugate space of Lp(x)(Ω) is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M16">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M17">View MathML</a>. For any u Lp(x)(Ω) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M18">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M19">View MathML</a>

Proposition 2. (see [1,5])If we denote ρ(u) = ∫|u|p(x)dx, ∀u Lp(x)(Ω), then

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

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M20">View MathML</a>

(iii) |u|p(x) → 0(∞) ⇔ ρ (u) → 0(∞).

From Proposition 2, the following inequalities hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M21">View MathML</a>

(3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M22">View MathML</a>

(4)

Proposition 3.If Ω ⊂ RN is a bounded domain, then the imbedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M23">View MathML</a>is compact whenever N < p-.

Proof. It is well know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M76">View MathML</a> is a continuous embedding, and the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M24">View MathML</a> is compact when N < p- and Ω is bounded. So we obtain the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M23">View MathML</a> is compact whenever N < p-.

From now on, we denote X by W1,p(x)(Ω) × W1,q(x)(Ω) with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M25">View MathML</a>

Then X is a separable and reflexive Banach spaces. Naturally, we denote X* by the space (W1,p(x))*(Ω) × (W1,q(x))*(Ω), the dual space of X.

From Proposition 3, we know that when p-,q- > N, the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M26">View MathML</a> is compact, there exist a positive constant c such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M27">View MathML</a>

(5)

3 Existence of three solutions

We define Φ, Ψ, J : X R by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M28">View MathML</a>

(6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M29">View MathML</a>

(7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M30">View MathML</a>

(8)

Then for any (ζ,η) ∈ X,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M31">View MathML</a>

We say that z = (u, v) ∈ X is a weak solution of problem (1) if for any (ζ, η) ∈ X

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M32">View MathML</a>

Thus, we deduce that z X is a weak solution of (1) if z is a solution of (2). It follows that we can seek for weak solutions of (1) by applying Theorem 1.

We first give the following result.

Lemma 1. If Φ is defined in (6), then (Φ')-1 : X* → X exists and it is continuous.

Proof. First, we show that Φ' is uniformly monotone. In fact, for any ζ, η RN, we have the following inequality (see [30]):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M33">View MathML</a>

Thus, we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M34">View MathML</a>

for any z1 = (u1, v1), z2 = (u2, v2) ∈ X, i.e.,Φ' is uniformly monotone.

From (3), (4), we can see that for any z X, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M35">View MathML</a>

That's meaning Φ' is coercive on X.

By a standard argument, we know that Φ' is hemicontinuous. Therefore, the conclusion follows immediately by applying Theorem 26.A [31].

To obtain our main result, we assume the following conditions on F(x,s,t):

(A1) There exist d(x) ∈ L1(Ω) and 0 < ς < p-, 0 < τ < q- such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M36">View MathML</a>

for a.e.x ∈ Ω and (s,t) ∈ R × R;

(A2) F(x,0,0) = 0 for a.e.x ∈ Ω;

(A3) There exist s1,t1 R with |s1|, |t1| ≥ 1 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M37">View MathML</a>

(9)

where c is given in (5) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M38">View MathML</a>

(A3)' F(x,s,t) > 0 for any x ∈ Ω and |s| or |t| large enough, and there exist M, N > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M39">View MathML</a>

Then we have the following main theorem.

Theorem 2. Assume (A1),(A2),(A3)(or (A3)'),(G) and (E) hold. Then there exist an open interval Λ ⊆ [0, ∞) and a positive real number ρ with the following property: for each λ ∈ Λ, there exists σ > 0 such that for each μ ∈ [0, σ], problem (1) has at least three weak solutions whose norms are less than ρ.

Proof. By the definitions of Φ, Ψ, J, we know that Ψ' is compact, Φ is weakly lower semi-continuous and bounded on each bounded subset of X. From lemma 1 we can see that (Φ')-1 is well defined, from condition (G), J is well defined and continuously Gâteaux differentiable on X, with compact derivative. Then we can use Theorem 1 to obtain the result. Now we show that the hypotheses of Theorem 1 are fulfilled.

Thanks to (A1), for each λ ≥ 0, one has that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M40">View MathML</a>

and so the assumption (i) of Theorem 1 holds.

Now we consider in two cases:

Case (i): (A3) holds, i.e., there exist 1 ≤ |s1|, |t1| such that (9) hold.

Now we set z0 = (0,0), z1 = (s1, s1) and denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M41">View MathML</a>, then it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M42">View MathML</a>

Thus, (ii) of Theorem 1 is satisfied.

At last, by (A2) we know Ψ(z0) = 0, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M43">View MathML</a>

(10)

On the other way, when Φ(z) ≤ r, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M44">View MathML</a>

We deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M45">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M46">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M47">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M48">View MathML</a>

By (5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M49">View MathML</a>

Thus, from (7), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M50">View MathML</a>

(11)

From (9)-(11) and the definition of r, we can see (iii) of Theorem 1 is hold.

Case (ii): (A3)' holds. Then there exist |s2|,|t2| > 1 such that F(x,s2,t2) > 0 for any x ∈ Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M51">View MathML</a>. Set a = min{c,M}, b = min{c, N} then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M52">View MathML</a>

(12)

We denote z2 = (s2,t2) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M53">View MathML</a>. Then it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M54">View MathML</a>

So, (ii) of Theorem 1 is satisfied.

When Φ(z) ≤ r, similar to the above arguments, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M55">View MathML</a>

(13)

At last, we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M56">View MathML</a>

(14)

From (7) and (12), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M57">View MathML</a>

(15)

From (14) and (15), we can see (iii) of Theorem 1 is still hold.

Then all the hypotheses of Theorem 1 are fulfilled. By Theorem 1, we know that there exist an open interval Λ ⊆ [0, ∞) and a positive constant ρ such that for any λ ∈ Λ, there exists σ > 0 and for each μ ∈ [0,σ], problem (1) has at least three weak solutions whose norms are less than ρ.

By Theorem 2, we have the following result.

Corollary 1. Let f, g : Ω × R R be Carathéodory functions, sup|ζ|≤s |g(·, ζ)| ∈ L1(Ω) for all s > 0, and define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M58">View MathML</a>for any (x,t) ∈ Ω × R, e(x) ∈ L(Ω) and ess infe(x) > 0. Assume the following conditions hold.

(B1) There exist d(x) ∈ L1(Ω) and 0 < ς < p- such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M59">View MathML</a>

for a.e.x ∈ Ω and t R;

(B2) There exists t3 R with |t3| ≥ 1 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M60">View MathML</a>

(16)

where c is given in (5) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M61">View MathML</a>

or

(B2)' F(x,t) > 0 for any x ∈ Ω and |t| large enough, and there exist M > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M62">View MathML</a>

Then there exist an open interval Λ ⊆ [0, ∞) and a positive constant ρ such that for any λ ∈ Λ, there exists σ > 0 and for each μ ∈ [0, σ], the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M63">View MathML</a>

(17)

has at least three weak solutions whose norms are less than ρ.

Remark 1. if p(x) = p in Ω, μ = 0, problem (17) was considered in [21]. If we take f(x,t) = |t|γ(x)-2t - t with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M64">View MathML</a> satisfies 2 < γ- γ+ < p-, μ = 0, Corollary 1 becomes a version of Theorem 2 in [23]. Hence our Corollary 1 unifies and generalizes Theorem 2 in [21] and Theorem 2 in [23] and our Theorem 2 generalizes the main results of [21-25] to the system (1).

At last, we give two examples.

Example 1. Let Ω = B(0,1) be the unit ball of RN with N ≥ 2, set p(x) = N + e|x|,q(x) = N + 1 + ln(1 + x2), ep(x) = (1 + x2) = eq(x), G(x,u,v) = x2(u2 + v2) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M65">View MathML</a>

(18)

where M is a positive constant, i.e., we consider the following problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M66">View MathML</a>

(19)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M67">View MathML</a>

(20)

We can see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M68">View MathML</a>, and it is easy to see that for any t1 > 1, there exists s1 > 1 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M69">View MathML</a>

(21)

were <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M70">View MathML</a> are positive constants and c is given by (5). Then when M s1, F(x,u,v) defined in (18) satisfies (A1)-(A3) of Theorem 2, and G(x,u,v),e(x) satisfy

(G) and (E) respectively, by Theorem 2, there exist an open interval Λ ⊆ [0, ∞) and a positive constant ρ such that for any λ ∈ Λ, there exists σ > 0 and for each μ ∈ [0,σ], system (19) has at least three weak solutions whose norms are less than ρ.

Example 2. Assume Ω,p(x),q(x),ep(x),eq(x),G(x,u,v) are the same as in example 1, and suppose N ≥ 8. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M71">View MathML</a>

(22)

Obviously, F(x,u,v) satisfies (A1) and (A2). By simple computation, we can see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M72">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M73">View MathML</a>

i.e., (A3)' hold for F(x,u,v) defined in (22).

Thus, there exist an open interval Λ ⊆ [0, ∞) and a positive constant ρ such that for any λ ∈ Λ, there exists σ > 0 and for each μ ∈ [0, σ], the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M74">View MathML</a>

(23)

has at least three weak solutions whose norms are less than ρ.

Remark 2. We remark that the methods used in this paper are also applicable for the cases of the other boundary value conditions, for example, Dirichlet boundary value conditions.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors read and approved the final manuscript.

Acknowledgements

The project supported by the National Natural Science Foundation of China (No. 11171092). Project supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 08KJB110005).

References

  1. Rúzicka, M: Electro-rheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Math, Springer-Verlag, Berlin (2000)

  2. Zhikov, VV: Averaging of functionals of the calculus of variations and elasticity theory. Math USSR Izv. 29, 33–66 (1987). Publisher Full Text OpenURL

  3. Acerbi, E, Mingione, G: Regularity results for a class of functionals with nonstandard growth. Arch Ration Mech Anal. 156, 121–140 (2001). Publisher Full Text OpenURL

  4. Fan, XL: Solutions for p(x)-Laplacian Dirichlet problems with singular coefficients. J Math Anal Appl. 312, 464–477 (2005). Publisher Full Text OpenURL

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

  6. Fan, XL, Wu, HQ, Wang, FZ: Hartman-type results for p(t)-Laplacian systems. Nonlinear Anal. 52, 585–594 (2003). Publisher Full Text OpenURL

  7. EI Hamidi, A: Existence results to elliptic systems with nonstandard growth con-ditions. J Math Anal Appl. 300, 30–42 (2004). Publisher Full Text OpenURL

  8. Zhang, QH: Existence and asymptotic behavior of positive solutions for variable exponent elliptic systems. Nonlinear Anal. 70, 305–316 (2009). Publisher Full Text OpenURL

  9. Zhang, QH, Qiu, ZM, Dong, R: Existence and asymptotic behavior of positive solutions for a variable exponent elliptic system without variational structure. Nonlinear Anal. 72, 354–363 (2010). Publisher Full Text OpenURL

  10. Alves, CO, de Morais Filho, DC, Souto, MAS: On systems of elliptic equations involving subcritical or critical Sobolev exponents. Nonlinear Anal. 42, 771–787 (2000). Publisher Full Text OpenURL

  11. Chen, CH: On positive weak solutions for a class of quasilinear elliptic systems. Nonlinear Anal. 62, 751–756 (2005). Publisher Full Text OpenURL

  12. Hai, DD, Shivaji, R: An existence result on positive solutions of p-Laplacian systems. Nonlinear Anal. 56, 1007–1010 (2004). Publisher Full Text OpenURL

  13. Hsu, TS: Multiple positive solutions for a critical quasilinear elliptic system with concave-convex non-linearities. Nonlinear Anal. 71, 2688–2698 (2009). Publisher Full Text OpenURL

  14. Han, P: The effect of the domain topology on the number of positive solutions of elliptic systems involving critical Sobolev exponents. Houston J Math. 32, 1241–1257 (2006)

  15. Kristály, A: Existence of two non-trivial solutions for a class of quasilinear elliptic variational systems on strip-like domains. Proc Edinb Math Soc II. 48, 465–477 (2005)

  16. Ricceri, B: On a three critical points theorem. Arch Math (Basel). 75, 220–226 (2000). Publisher Full Text OpenURL

  17. Wu, TF: The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions. Nonlinear Anal. 68, 1733–1745 (2008). Publisher Full Text OpenURL

  18. Yin, HH, Yang, ZD: Existence and nonexistence of entire positive solutions for quasilinear systems with singular and super-linear terms. Diff Equ Appl. 2(2), 241–249 (2010)

  19. Yin, HH, Yang, ZD: Multiplicity results for a class of concave-convex elliptic systems involving sign-changing weight. Ann Pol Math. 102(1), 51–71 (2011). Publisher Full Text OpenURL

  20. Afrouzi, GA, Heidarkhani, S: Three solutions for a Dirichlet boundary value problem involving the p-Laplacian. Nonlinear Anal. 66, 2281–2288 (2007). Publisher Full Text OpenURL

  21. Bonanno, G, Candito, P: Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian. Arch Math (Basel). 80, 424–429 (2003)

  22. Candito, P: Existence of three solutions for a nonautonomous two point boundary value problem. J Math Anal Appl. 252, 532–537 (2000). Publisher Full Text OpenURL

  23. Mihăilescu, M: Existence and multiplicity of solutions for a Neumann problem involving the p(x)-Laplace operator. Nonlinear Anal. 67, 1419–1425 (2007). Publisher Full Text OpenURL

  24. Li, C, Tang, CL: Three solutions for a class of quasilinear elliptic systems involving the (p,q)-Laplacian. Nonlinear Anal. 69, 3322–3329 (2008). Publisher Full Text OpenURL

  25. El Manouni, S, Kbiri Alaoui, M: A result on elliptic systems with Neumann conditions via Ricceri's three critical points theorem. Nonlinear Anal. 71, 2343–2348 (2009). Publisher Full Text OpenURL

  26. Ricceri, B: A three critical points theorem revisited. Nonlinear Anal. 70, 3084–3089 (2009). Publisher Full Text OpenURL

  27. Bonanno, G: A minimax inequality and its applications to ordinary differential equations. J Math Anal Appl. 270, 210–229 (2002). Publisher Full Text OpenURL

  28. Kováčik, O, Rákosník, J: On the spaces Lp(x)(Ω) and Wk,p(x)(Ω). Czechoslovak Math J. 41, 592–618 (1991)

  29. Sanko, SG: Denseness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/30/mathml/M75">View MathML</a> in the generalized Sobolev spaces Wm,p(x)(RN). Dokl Ross Akad Nauk. 369(4), 451–454 (1999)

  30. Kichenassamy, S, Veron, L: Singular solutions of the p-Laplace equation. Math Ann. 275, 599–615 (1985)

  31. Zeider, E: Nonlinear Functional Analysis and its Applications, II/B: Nonlinear Monotone Operators. Springer, New York (1990)