Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 65, 1-13; http://www.math.u-szeged.hu/ejqtde/

Existence of solutions for p(x)−Laplacian equations^{1}

R. A. Mashiyev^{a,}^{2}, B. Cekic^{a} and O. M. Buhrii^{b}

aDepartment of Mathematics, Dicle University 21280-Diyarbakir, Turkey. bDepartment of Differential Equations, Ivan Franko National University of Lviv,

79000-Lviv, Ukraine mrabil@dicle.edu.tr & bilalc@dicle.edu.tr & ol buhrii@i.ua

Abstract

We discuss the problem

\{ \begin{matrix}− div (|∇u|^{p(x)−2} ∇u) = λ(a (x) |u|^{q(x)−2} u + b(x) |u|^{h(x)−2} u), & for x ∈ Ω, \\ u = 0, & for x ∈ ∂Ω. \end{matrix}

where Ω is a bounded domain with smooth boundary in ℝ^{N} (N ≥ 2) and p is Lipschitz continuous, q and h are continuous functions on \overline{Ω} such that 1 < q(x) < p(x) < h(x) < p^{∗}(x) and p(x) < N. We show the existence of at least one nontrivial weak solution. Our approach relies on the variable exponent theory of Lebesgue and Sobolev spaces combined with adequate variational methods and the Mountain Pass Theorem.

Keywords and Phrases: variable exponent Lebesgue and Sobolev spaces; p (x)-Laplacian; variational methods; Mountain Pass Theorem.

AMS Subject Classifications (2000): 35J60, 35B30, 35B40

1. Introduction

The study of partial differential equations and variational problems involving p(x)-growth conditions has captured special attention in the last decades. This is a consequence of the fact that such equations can be used to model phenomena which arise in mathematical physics, for example:

• Electrorheological fluids: see Acebri and Mingione [1], Zhikov [25] and Růžička [20], Fan and Zhang [12], Mihăilescu and Rădulescu [16], Chabrowski and Fu [7], Hästö [14], Diening [8].

• Nonlinear porous medium: see Antontsev and Rodrigues [2], Buhrii and Mashiyev [4], and Songzhe, Wenjie, Chunling and Hongjun [21].

• Image Processing: Chen, Levine and Rao [6].

A typical model of an elliptic equation with p(x)-growth conditions is

− div (|∇u|^{p(x)−2} ∇u) = f(x, u).  (1.1)

The operator − div (|∇u|^{p(x)−2} ∇u) is called the p(x)-Laplace operator and it is a natural generalization of the p−Laplace operator, in which p(x) ≡ p > 1 is a constant. The p(x)-Laplacian processes have more complicated nonlinearity, for example, it is nonhomogeneous, so in the discussions some special techniques will be needed.

1This research was supported by DUBAP grant No. 10-FF-15, Dicle University, Turkey.

2Corresponding author, E-mail: mrabil@dicle.edu.tr

Problems like (1.1) with Dirichlet boundary condition have been largely considered in the literature in the recent years. We give in what follows a concise but complete image of the actual stage of research on this topic. We will use the notations such as p_{1} and p_{2} where

p_{1} := ess inf_{x∈Ω} p (x) ≤ p (x) ≤ p_{2} := ess sup_{x∈Ω} p (x) < ∞.

In the case f(x, u) = λ |u|^{p(x)−2} u in [13] the authors established the existence of infinitely many eigenvalues for problem (1.1) by using an argument based on the Ljusternik-Schnirelmann critical point theory. Denoting by Λ the set of all nonnegative eigenvalues, they showed that Λ is discrete, sup Λ = ∞ and pointed out inf Λ = 0 for general p(x), and only under some special conditions inf Λ > 0. In the case f(x, u) = λ |u|^{q(x)−2} u, there are different papers, for example, in [12] the same authors proved that any λ > 0 is an eigenvalue of problem (1.1) when p_{2} < q_{1} and also when q_{2} < p_{1}. In [18] the authors proved the existence of a continuous family of eigenvalues which lies in a neighborhood of the origin when q_{1} < p_{1} and q(x) has subcritical growth in problem (1.1).

In the case f(x, u) = A |u|^{a−2} u + B |u|^{b−2} u with 1 < a < p_{1} < p_{2} < b < min\{N, \frac{Np_{1}}{N−p_{1}} \} and A, B > 0, in [17] Mihăilescu show that there exists λ > 0 such that, for any A, B ∈ (0, λ), problem (1.1) has at least two distinct nontrivial weak solutions.

The aim of this paper is to discuss the existence of a weak solution of the p (x)-Laplacian equation

\{ \begin{matrix}− div (|∇u|^{p(x)−2} ∇u) = λ(a (x) |u|^{q(x)−2} u + b(x) |u|^{h(x)−2} u), & for x ∈ Ω, \\ u = 0, & for x ∈ ∂Ω. \end{matrix}  (P_{λ})

where Ω ⊂ ℝ^{N} (N ≥ 2) is a bounded domain with smooth boundary, λ is a positive real number, p is Lipschitz continuous on \overline{Ω}, and q, h ∈ C_{+}(\overline{Ω}), a(x), b(x) > 0 for x ∈ \overline{Ω} such that a ∈ L^{β(x)} (Ω), β(x) = \frac{p(x)}{p(x)−q(x)} and b ∈ L^{γ(x)} (Ω), γ(x) = \frac{p^{∗}(x)}{p^{∗}(x)−h(x)}. Here p^{∗}(x) = \frac{Np(x)}{N−p(x)} if p(x) < N or p^{∗}(x) = ∞ if p(x) ≥ N.

In the present paper, assuming the condition

1 < q_{1} ≤ q_{2} < p_{1} ≤ p_{2} < h_{1} ≤ h_{2} < p_{1}^{∗} and p_{2} < N  (1.2)

and using the the variable exponent theory of Lebesgue and Sobolev spaces combined with adequate variational methods and the Mountain Pass Theorem, we show the existence of at least one nontrivial weak solution of problem (P_{λ}).

2.Preliminaries

We recall in what follows some definitions and basic properties of variable exponent Lebesgue and Sobolev spaces L^{p(x)} (Ω), W^{1,p(x)} (Ω) and W_{0}^{1,p(x)} (Ω). In that context we refer to [9, 10, 15] for the fundamental properties of these spaces.

Set

L_{+}^{∞} (Ω) = {p; p ∈ L^{∞} (Ω) , ess inf_{x∈Ω} p(x) > 1 } .

For p ∈ L_{+}^{∞} (Ω) , we define the variable exponent Lebesgue space L^{p(.)}(Ω) to consist of all measurable functions u : Ω → ℝ for which the modular

ρ_{p(x)} (u) = ∫_{Ω} |u (x)|^{p(x)} dx

is finite. We define the Luxembourg norm on this space by the formula

∥u∥_{p(x)} = inf \{δ > 0 : ρ_{p(x)} (\frac{u}{δ}) ≤ 1\} .

Equipped with this norm, L^{p(.)}(Ω) is a separable and reflexive Banach space. Define the variable exponent Sobolev space W^{1,p(x)} (Ω) by

W^{1,p(x)} (Ω) = {u ∈ L^{p(x)} (Ω) ; |∇u| ∈ L^{p(x)} (Ω)},

and the norm

∥u∥_{1,p(x)} = ∥u∥_{p(x)} + ∥∇u∥_{p(x)} , ∀u ∈ W^{1,p(x)} (Ω)

makes W^{1,p(x)} (Ω) a separable and reflexive Banach space. The space W_{0}^{1,p(x)} (Ω) is denoted by the closure of C_{0}^{∞}(Ω) in W^{1,p(x)} (Ω). W_{0}^{1,p(x)} (Ω) is a separable and reflexive Banach space.

Proposition 2.1 [10, 15] The conjugate space of L^{p(x)} (Ω) is L^{q(x)} (Ω), where \frac{1}{p(x)} + \frac{1}{q(x)} = 1. For any u ∈ L^{p(x)} (Ω) and υ ∈ L^{q(x)} (Ω), we have

|∫_{Ω} uυdx| ≤ (\frac{1}{p_{1}} + \frac{1}{q_{1}} ) ∥u∥_{p(x)} ∥υ∥_{q(x)} ≤ 2 ∥u∥_{p(x)} ∥υ∥_{q(x)} .

The next proposition illuminates the close relation between the ∥·∥_{p(x)} and the convex modular ρ_{p(x)} :

Proposition 2.2 [9, 10, 15] If u ∈ L^{p(x)} (Ω) and p_{2} < ∞ then we have

i) ∥u∥_{p(x)} < 1 (= 1; > 1) ⇔ ρ_{p(x)}(u) < 1 (= 1; > 1) ,
ii) ∥u∥_{p(x)} > 1 ⟹ ∥u∥_{p(x)}^{p_{1}} ≤ ρ_{p(x)} (u) ≤ ∥u∥_{p(x)}^{p_{2}} ,
iii) ∥u∥_{p(x)} < 1 ⟹ ∥u∥_{p(x)}^{p_{2}} ≤ ρ_{p(x)} (u) ≤ ∥u∥_{p(x)}^{p_{1}} ,
iv) ∥u∥_{p(x)} = a > 0 ⟺ ρ_{p(x)} (\frac{u}{a}) = 1

Proposition 2.3 [9, 10, 15] If u, u_{n} ∈ L^{p(x)} (Ω) , n = 1, 2, ..., then the following statements are equivalent

(1) lim_{n→∞} ∥u_{n} − u∥_{p(x)} = 0;

(2) lim_{n→∞} ρ_{p(x)}(u_{n} − u) = 0;

(3) u_{n} → u in measure in Ω and lim_{n→∞} ρ_{p(x)}(u_{n}) = ρ_{p(x)}(u).

Lemma 2.4 [5] Assume that r ∈ L_{+}^{∞} (Ω) and p ∈ C_{+}(\overline{Ω}) := \{m ∈ C(\overline{Ω}) : m_{1} > 1\} . If |u|^{r(x)} ∈ L^{p(x)} (Ω), then we have

min {∥u∥_{r(x)p(x)}^{r_{1}} , ∥u∥_{r(x)p(x)}^{r_{2}}} ≤ ∥|u|^{r(x)}∥_{p(x)} ≤ max {∥u∥_{r(x)p(x)}^{r_{1}} , ∥u∥_{r(x)p(x)}^{r_{2}}} .

Remark 2.5 If r(x) ≡ r, r ∈ ℝ then

∥|u|^{r}∥_{p(x)} = ∥u∥_{rp(x)}^{r} .

Given two Banach spaces X and Y, the symbol X ↪ Y means that X is continuously imbedded in Y and the symbol X ↪↪ Y means that there is a compact embedding of X in Y.

Proposition 2.6 [9, 10, 11, 15] Assume that Ω is bounded and smooth.

(i) Let q, h ∈ C_{+}(\overline{Ω}). If q(x) ≤ h(x) for all x ∈ \overline{Ω}, then L^{h(x)} (Ω) ↪ L^{q(x)} (Ω) . (ii) Let p is Lipschitz continuous and p_{2} < N, then for h ∈ L_{+}^{∞}(Ω) with p(x) ≤ h(x) ≤ p^{∗}(x)

there is a continuous imbedding W^{1,p(x)} (Ω) ↪ L^{h(x)} (Ω) , and also there is a constant C_{1} > 0 such that ∥u∥_{h(x)} ≤ C_{1} ∥u∥_{1,p(x)}.

(iii) Let p, q ∈ C_{+}(\overline{Ω}). If p(x) ≤ q(x) < p^{∗}(x) for all x ∈ \overline{Ω}, then W^{1,p(x)} (Ω) ↪↪ L^{q(x)} (Ω) . (iv) (Poincaré inequality ) If p ∈ C_{+}(\overline{Ω}), then there is a constant C_{2} > 0 such that

∥u∥_{p(x)} ≤ C_{2} ∥|∇u|∥_{p(x)} , ∀ u ∈ W_{0}^{1,p(x)} (Ω) .

Consequently, ∥u∥ := ∥|∇u|∥_{p(x)} and ∥u∥_{1,p(x)} are equivalent norms on W_{0}^{1,p(x)} (Ω). In what fol-

lows, W_{0}^{1,p(x)} (Ω), with p ∈ C_{+}(\overline{Ω}), will be considered as endowed with the norm ∥u∥_{1,p(x)}. We will

use ∥u∥ = ∥∇u∥_{p(x)} for u ∈ W_{0}^{1,p(x)} (Ω) in the following discussions.

Finally, we introduce Mountain-Pass Theorem which is the main tool of the present paper.

Palais-Smale condition [24] Let E be a Banach space and I ∈ C^{1} (E, ℝ). If {u_{n}} ⊂ E is a sequence which satisfies conditions

|I_{λ}(u_{n})| < M,
I′_{λ}(u_{n}) → 0 as n → ∞ in E^{∗}

where M is a positive constant and E^{∗} is the dual space of E, then {u_{n}} possesses a convergent subsequence.

Mountain-Pass Theorem [24] Let E be a Banach space, and let I ∈ C^{1} (E, ℝ) satisfy the Palais-Smale condition. Assume that I (0) = 0, and there exists a positive real number ρ and u, υ ∈ E such that

(i) ∥υ∥ > ρ, I (υ) ≤ I (0) . (ii) α = inf {I (u) : u ∈ E, ∥u∥ = ρ} > 0. Put G = {g ∈ C ([0, 1] , E) : g (0) = 0, g (1) = υ} ≠ ∅. Set β = inf_{g∈G} sup_{t∈[0,1]}I (g (t)) .

Then, β ≥ α and β is a critical value of I.

3. Main Results

The energy functional corresponding to problem (P_{λ}) is defined as J_{λ} : W_{0}^{1,p(x)} (Ω) → ℝ,

J_{λ}(u) = ∫_{Ω} \frac{|∇u|^{p(x)}}{p(x)} dx − λ ∫_{Ω} \frac{a(x)}{q(x)} |u|^{q(x)} dx − λ ∫_{Ω} \frac{b(x)}{h(x)} |u|^{h(x)} dx.  (3.1)

We say that u ∈ W_{0}^{1,p(x)} (Ω) is a weak solution for problem (P_{λ}) provided

∫_{Ω} |∇u|^{p(x)−2} ∇u∇υdx = λ ∫_{Ω} a(x) |u|^{q(x)−2} uυdx + λ ∫_{Ω} b(x) |u|^{h(x)−2} uυdx

for all υ ∈ W_{0}^{1,p(x)} (Ω) .

Standard arguments imply that J_{λ} ∈ C^{1} (W_{0}^{1,p(x)} (Ω) , ℝ) with

⟨J′_{λ}(u), υ⟩ = ∫_{Ω} |∇u|^{p(x)−2} ∇u∇υdx − λ ∫_{Ω} a(x) |u|^{q(x)−2} uυdx − λ ∫_{Ω} b(x) |u|^{h(x)−2} uυdx (3.2)

for all u, υ ∈ W_{0}^{1,p(x)} (Ω). Thus the weak solution of (P_{λ}) are exactly the critical points of J_{λ}. The main result of the present paper is the following theorem. Theorem 3.1 Assume p is Lipschitz continuous, q, h ∈ C_{+}(\overline{Ω}) and condition (1.2) is fulfilled.

If

λ ∈ (0, min \{\frac{q_{1}ρ^{p_{2}−q_{1}}}{4C_{1}^{q_{1}}p_{2} ∥a∥_{β(x)}} , \frac{h_{1}ρ^{p_{2}−h_{1}}}{4C_{1}^{h_{1}}p_{2} ∥b∥_{γ(x)}} \}) ,

then the problem (P_{λ}) has at least one nontrivial solution, where ρ ∈ (0, 1) .

To obtain the proof of Theorem 3.1, we use Mountain-Pass theorem. Therefore, we must show J_{λ} satisfies Palais-Smale condition in the first place.

Lemma 3.2 Let λ satisfies the condition of Theorem 3.1. If {u_{n}} ⊂ W_{0}^{1,p(x)} (Ω) is a sequence which satisfies conditions

|J_{λ}(u_{n})| < M,  (3.3)
J′_{λ}(u_{n}) → 0 as n → ∞ in (W_{0}^{1,p(x)} (Ω))^{∗}  (3.4)

where M is a positive constant, then {u_{n}} possesses a convergent subsequence.

Proof: First, we show that {u_{n}} is bounded in W_{0}^{1,p(x)} (Ω). Assume the contrary. Then, passing to a subsequence if necessary, we may assume that ∥u_{n}∥ → ∞ as n → ∞. Thus, we may consider

EJQTDE, 2010 No. 65, p. 5 that ∥u_{n}∥ > 1 for any integer n. By (3.4) we deduce that there exists N_{1} > 0 such that for any n > N_{1}, we have

∥J′_{λ}(u_{n})∥ ≤ 1.

On the other hand, for any n > N_{1} fixed, the application

W_{0}^{1,p(x)} (Ω) ∋ υ → ⟨J′_{λ}(u_{n}), υ⟩

is linear and continuous. The above information implies

|⟨J′_{λ}(u_{n}), υ⟩| ≤ ∥J′_{λ}(u_{n})∥_{W_{0}^{−1,p′(x)}(Ω)} ∥υ∥ ≤ ∥υ∥, ∀υ ∈ W_{0}^{1,p(x)} (Ω) , n > N_{1}.

Setting υ = u_{n} we have

− ∥u_{n}∥ ≤ ∫_{Ω} |∇u_{n}|^{p(x)} dx − λ ∫_{Ω} a(x) |u_{n}|^{q(x)} dx − λ ∫_{Ω} b(x) |u_{n}|^{h(x)} dx ≤ ∥u_{n}∥

for any n > N_{1}.

Using the assumption ∥u_{n}∥ > 1, relations (3.3) , (3.4), Proposition 2.1, Lemma 2.4 and Proposition 2.6 (ii) we have

M > J_{λ}(u_{n}) − \frac{1}{h_{1}} ⟨J′_{λ}(u_{n}), u_{n}⟩
= ∫_{Ω} \frac{|∇u_{n}|^{p(x)}}{p(x)} dx − λ ∫_{Ω} \frac{a(x)}{q(x)} |u_{n}|^{q(x)} dx − λ ∫_{Ω} \frac{b(x)}{h(x)} |u_{n}|^{h(x)} dx

−\frac{1}{h_{1}} ∫_{Ω} |∇u_{n}|^{p(x)} dx + \frac{λ}{h_{1}} ∫_{Ω} a(x) |u_{n}|^{q(x)} dx + \frac{λ}{h_{1}} ∫_{Ω} b(x) |u_{n}|^{h(x)} dx

≥ \frac{1}{p_{2}} ∫_{Ω} |∇u_{n}|^{p(x)} dx − \frac{λ}{q_{1}} ∫_{Ω} a(x) |u_{n}|^{q(x)} dx − \frac{λ}{h_{1}} ∫_{Ω} b(x) |u_{n}|^{h(x)} dx

−\frac{1}{h_{1}} ∫_{Ω} |∇u_{n}|^{p(x)} dx + \frac{λ}{h_{1}} ∫_{Ω} a(x) |u_{n}|^{q(x)} dx + \frac{λ}{h_{1}} ∫_{Ω} b(x) |u_{n}|^{h(x)} dx

≥ (\frac{1}{p_{2}} − \frac{1}{h_{1}} ) ∥u_{n}∥^{p_{1}} − λ (\frac{1}{h_{1}} − \frac{1}{q_{1}} ) C_{3} ∥a∥_{β(x)} ∥u_{n}∥^{q_{2}} ,  (3.5)

where C_{3} > 0 is a constant independent of u_{n} and x, for n large enough. Dividing (3.5) by ∥u_{n}∥^{p_{1}} and passing to the limit as n → ∞ we obtain \frac{1}{p_{2}} − \frac{1}{h_{1}} < 0.

Since q_{1} ≤ q_{2} < p_{1} ≤ p_{2} < h_{1}, this is a contradiction. It follows {u_{n}} is bounded in W_{0}^{1,p(x)} (Ω) .) Next, we show the strong convergence of {u_{n}} in W_{0}^{1,p(x)} (Ω). Since {u_{n}} is bounded, up to a

subsequence (which we still denote by {u_{n}}), we may assume that there exists u ∈ W_{0}^{1,p(x)} (Ω) such that

u_{n} ⇀ u weakly in W_{0}^{1,p(x)} (Ω) as n → ∞.

By Proposition 2.6 (iii) we obtain

u_{n} → u strongly in L^{p(x)} (Ω) as n → ∞.  (3.6)

Furthermore, from [3, 23] we have

u_{n} → u strongly in L^{p^{∗}(x)} (K) as n → ∞,  (3.7)

where K is compact subset of Ω.

The above information and relation (3.4) imply

⟨J′_{λ}(u_{n}) − J′_{λ}(u), u_{n} − u⟩ → 0 as n → ∞.

On the other hand, we have

∫_{Ω} (|∇u_{n}|^{p(x)−2} ∇u_{n} − |∇u|^{p(x)−2} ∇u) (∇u_{n} − ∇u) dx

= ⟨J′_{λ}(u_{n}) − J′_{λ}(u), u_{n} − u⟩ − λ ∫_{Ω} a(x) (|u_{n}|^{q(x)−2} u_{n} − |u|^{q(x)−2} u) (u_{n} − u) dx

−λ ∫_{Ω} b(x) (|u_{n}|^{h(x)−2} u_{n} − |u|^{h(x)−2} u) (u_{n} − u) dx.

Propositions 2.1, 2.3 and Lemma 2.4 we have

λ |∫_{Ω} a(x) (|u_{n}|^{q(x)−2} u_{n} − |u|^{q(x)−2} u_{1}) (u_{n} − u) dx|

≤ λ |∫_{Ω} a(x) |u_{n}|^{q(x)−2} u_{n} (u_{n} − u) dx| + λ |∫_{Ω} a(x) |u_{1}|^{q(x)−2} u (u_{n} − u) dx|

≤ C_{4} ∥a∥_{β(x)} ∥|u_{n}|^{q(x)−1}∥_{\frac{p(x)}{q(x)−1}} ∥u_{n} − u∥_{p(x)} + C_{5} ∥a∥_{β(x)} |||u|^{q(x)−1}||_{\frac{p(x)}{q(x)−1}} ∥u_{n} − u∥_{p(x)}
≤ C_{4} ∥a∥_{β(x)} ∥u_{n}∥_{p(x)}^{q_{2}−1} ∥u_{n} − u∥_{p(x)} + C_{5} ∥a∥_{β(x)} ∥u∥_{p(x)}^{q_{2}−1} ∥u_{n} − u∥_{p(x)} ,

where C_{4}, C_{5} > 0 and \frac{1}{β(x)} + \frac{q(x)−1}{p(x)} + \frac{1}{p(x)} = 1.

Similarly, Propositions 2.1, 2.3 and Lemma 2.4 we have

λ |∫_{Ω} b(x) (|u_{n}|^{h(x)−2} u_{n} − |u|^{h(x)−2} u) (u_{n} − u) dx|

≤ C_{6} ∥b∥_{γ(x)} ∥|u_{n}|^{h(x)−1}∥_{\frac{p^{∗}(x)}{h(x)−1}} ∥u_{n} − u∥_{p^{∗}(x)} + C_{7} ∥b∥_{γ(x)} ∥|u|^{h(x)−1}∥_{\frac{p^{∗}(x)}{h(x)−1}} ∥u_{n} − u∥_{p^{∗}(x)}
≤ C_{6} ∥b∥_{γ(x)} ||u_{n}||_{p^{∗}(x)}^{h_{2}−1} ∥u_{n} − u∥_{p^{∗}(x)} + C_{7} ∥b∥_{γ(x)} ∥u∥_{p^{∗}(x)}^{h_{2}−1} ∥u_{n} − u∥_{p^{∗}(x)} ,

where C_{6}, C_{7} > 0 and \frac{1}{γ(x)} + \frac{h(x)−1}{p^{∗}(x)} + \frac{1}{p^{∗}(x)} = 1. By (3.6) and (3.7) we have

∥u_{n} − u∥_{p(x)} → 0 as n → ∞

and

∥u_{n} − u∥_{p^{∗}(x)} → 0 as n → ∞.

Therefore, from above inequalities we deduce that

lim_{n→∞} ∫_{Ω} b(x) (|u_{n}|^{h(x)−2} u_{n} − |u|^{h(x)−2} u) (u_{n} − u) = 0,  (3.8)

and

lim_{n→∞} ∫_{Ω} a(x) (|u_{n}|^{q(x)−2} u_{n} − |u|^{q(x)−2} u) (u_{n} − u) dx = 0,  (3.9)

respectively. By (3.8) and (3.9) we obtain

lim_{n→∞} ∫_{Ω} (|∇u_{n}|^{p(x)−2} ∇u_{n} − |∇u|^{p(x)−2} ∇u) (∇u_{n} − ∇u) = 0.  (3.10)

This result and the following inequality [23, Lemma 2.2]

(|ξ|^{r−2} ξ − |η|^{r−2} η) (ξ − η) ≥ 2^{−r} |ξ − η|, ∀r ≥ 2; ξ, η ∈ ℝ^{N}.  (3.11)

yield

lim_{n→∞} ∫_{Ω} |∇u_{n} − ∇u|^{p(x)} dx = 0.  (3.12)

This fact and Proposition 2.3 imply ∥∇u_{n} − ∇u∥_{p(x)} → 0 as n → ∞. Relation (3.12) and fact that u_{n} ⇀ u (weakly) in W_{0}^{1,p(x)} (Ω) enable us to apply [12] in order to obtain that u_{n} ⇀ u (strongly) in W_{0}^{1,p(x)} (Ω). Thus, Lemma 3.2 is proved.

Now, we show that the Mountain-Pass theorem can be applied in this case. Lemma 3.3 Assume p, q, h ∈ C_{+}(\overline{Ω}) and condition (1.2) is fulfilled. The following assertions

hold.

(i) There exist λ > 0, α > 0 and ρ ∈ (0, 1) such that

J_{λ}(u) ≥ α, ∀u ∈ W_{0}^{1,p(x)} (Ω) with ∥u∥ = ρ.  (3.13)

(ii) There exists ω ∈ W_{0}^{1,p(x)} (Ω) such that

lim_{t→∞}J_{λ}(tω) = −∞.  (3.14)

(iii) There exists φ ∈ W_{0}^{1,p(x)} (Ω) such that φ ≥ 0, φ ≠ 0 and

J_{λ}(tφ) < 0,  (3.15)

EJQTDE, 2010 No. 65, p. 8 for t > 0 small enough. Proof. (i) Using Propositions 2.1, 2.2, 2.6 (ii) and Lemma 2.4 we deduce that for any u ∈

W_{0}^{1,p(x)} (Ω) with ρ ∈ (0, 1) we have

J_{λ}(u) ≥ \frac{1}{p_{2}} ∫_{Ω} |∇u|^{p(x)} dx − \frac{λ}{q_{1}} ∫_{Ω} a(x) |u|^{q(x)} dx − \frac{λ}{h_{1}} ∫_{Ω} b(x) |u|^{h(x)} dx
≥ \frac{1}{p_{2}} ∥u∥^{p_{2}} − \frac{λ}{q_{1}} ∥a∥_{β(x)} ∥|u|^{q(x)}∥_{\frac{p(x)}{q(x)}} − \frac{λ}{h_{1}} ∥b∥_{γ(x)} ∥|u|^{h(x)}∥_{\frac{p^{∗}(x)}{h(x)}}
≥ \frac{1}{p_{2}} ∥u∥^{p_{2}} − \frac{λ}{q_{1}} ∥a∥_{β(x)} ∥u∥_{p(x)}^{q_{1}} − \frac{λ}{h_{1}} ∥b∥_{γ(x)} ∥u∥_{p^{∗}(x)}^{h_{1}}
≥ \frac{1}{p_{2}} ∥u∥^{p_{2}} − C_{1}^{q_{1}} \frac{λ}{q_{1}} ∥a∥_{β(x)} ∥u∥^{q_{1}} − C_{1}^{h_{1}} \frac{λ}{h_{1}} ∥b∥_{γ(x)} ∥u∥^{h_{1}} .

Taking

λ = min \{\frac{q_{1}ρ^{p_{2}−q_{1}}}{4C_{1}^{q_{1}}p_{2} ∥a∥_{β(x)}} , \frac{h_{1}ρ^{p_{2}−h_{1}}}{4C_{1}^{h_{1}}p_{2} ∥b∥_{γ(x)}} \}

we obtain

J_{λ}(u) ≥ \frac{ρ^{p_{2}}}{2p_{2}} , ∀u ∈ W_{0}^{1,p(x)} (Ω) with ∥u∥ = ρ.

Thus Lemma 3.3 (i) is proved.

(ii) Let ω ∈ C_{0}^{∞} (Ω) , ω ≥ 0, ω ≠ 0 and t > 1.We have

J_{λ}(tω) = ∫_{Ω} \frac{t^{p(x)}}{p(x)} |∇ω|^{p(x)} dx − λ ∫_{Ω} \frac{t^{q(x)}}{q(x)} a(x) |ω|^{q(x)} dx − λ ∫_{Ω} \frac{t^{h(x)}}{h(x)} b(x) |ω|^{h(x)} dx
≤ \frac{t^{p_{2}}}{p_{1}} ∫_{Ω} |∇ω|^{p(x)} dx − λ\frac{t^{q_{2}}}{q_{1}} ∫_{Ω} a(x) |ω|^{q(x)} dx − λ\frac{t^{h_{2}}}{h_{1}} ∫_{Ω} b(x) |ω|^{h(x)} dx.

Since q_{2}, p_{2} < h_{2} we have J_{λ}(tω) → −∞. Thus Lemma 3.3 (ii) is proved.

(iii) Let φ ∈ C_{0}^{∞} (Ω) , φ ≥ 0, φ ≠ 0 and t ∈ (0, 1). We have

J_{λ}(tφ) ≤ \frac{t^{p_{1}}}{p_{1}} ∫_{Ω} |∇φ|^{p(x)} dx − λ\frac{t^{q_{1}}}{q_{1}} ∫_{Ω} a(x) |φ|^{q(x)} dx − λ\frac{t^{h_{1}}}{h_{1}} ∫_{Ω} b(x) |φ|^{h(x)} dx.

Since

\frac{λt^{q_{1}}}{q_{1}} ∫_{Ω} a(x) |φ|^{q(x)} dx + \frac{λt^{h_{1}}}{h_{1}} ∫_{Ω} b(x) |φ|^{h(x)} dx

< \frac{λt^{q_{1}}}{q_{1}} (∫_{Ω} a(x) |φ|^{q(x)} dx + ∫_{Ω} b(x) |φ|^{h(x)} dx) ,

we have

J_{λ}(tφ) ≤ \frac{t^{p_{1}}}{p_{1}} ∫_{Ω} |∇φ|^{p(x)} dx − \frac{λt^{q_{1}}}{q_{1}} (∫_{Ω} a(x) |φ|^{q(x)} dx + ∫_{Ω} b(x) |φ|^{h(x)} dx) < 0,

for t < δ^{\frac{1}{p_{1}−q_{1}}} with

0 < δ < min \{1, \frac{λp_{1} (∫_{Ω} a(x) |φ|^{q(x)} dx + ∫_{Ω} b(x) |φ|^{h(x)} dx)}{q_{1} ∫_{Ω} |∇φ|^{p(x)} dx}\} .

Lemma 3.3 (iii) is proved.

Proof of Theorem 3.1 We set

G = {g ∈ C([0, 1] , W_{0}^{1,p(x)} (Ω)) : g (0) = 0, g (1) = υ} ,

where υ ∈ W_{0}^{1,p(x)} (Ω) is determined by Lemma 3.3 (ii) and (iii), and

β := inf_{g∈G} sup_{t∈[0,1]}J_{λ} (g (t)) .

According to Lemma 3.3 (ii) and (iii), we know that ∥υ∥ > ρ, so every path g ∈ G intersects the sphere ∥v∥ = ρ. Then Lemma 3.3 (i) implies

α ≤ inf_{∥u∥=ρ}J_{λ} (u) ≤ β,

with the constant α > 0 in Lemma 3.3 (i), thus β > 0. By the Mountain-Pass theorem J_{λ} admits a critical value β ≥ α.

Since J_{λ} ∈ C^{1} (W_{0}^{1,p(x)} (Ω) , ℝ), from Lemma 3.2 we conclude

J′_{λ}(u_{n}) → J′_{λ}(u)  (3.16)

as n → ∞.

Relations (3.3), (3, 4) and (3.16) show that J′_{λ}(u) = 0 and thus u is a weak solutions for problem (P_{λ}). Moreover, by relations (3.3), (3, 4) it follows that J_{λ}(u) > 0 and thus, u is a nontrivial weak solutions for problem (P_{λ}). The proof is completed.

Remark 3.4 If (u, λ) is a solution of (P_{λ}) and u ≠ 0, as usual, we call λ and u an eigenvalue eigenfunction corresponding to λ of (P_{λ}), respectively. If (u, λ) is a solution of (P_{λ}) and u ≠ 0, then

λ = λ(u) = \frac{∫_{Ω} |∇u|^{p(x)} dx}{∫_{Ω} (a(x) |u|^{q(x)} + b(x) |u|^{h(x)})dx}

and hence λ > 0.

Theorem 3.1 ensures that problem (P_{λ}) has a continuous family of positive eigenvalues that lie in a neighborhood of the origin. Furthermore, we obtain

inf_{u∈W_{0}^{1,p(x)}\\{0\}} \frac{∫_{Ω} |∇u|^{p(x)} dx}{∫_{Ω} (a(x) |u|^{q(x)} + b(x) |u|^{h(x)})dx} = 0.

Remark 3.5 Furthermore, we can consider the equation

\{ \begin{matrix}− div (|∇u|^{p(x)−2} ∇u) = λ(a (x) |u|^{q(x)−2} u − b(x) |u|^{h(x)−2} u), & for x ∈ Ω, \\ u = 0, & for x ∈ ∂Ω. \end{matrix}  (P_{λ}^{/})

where p, q, h, a, b and λ are the same in problem (P_{λ}).

The energy functional corresponding to problem (P_{λ}^{/}) is defined as I_{λ} : W_{0}^{1,p(x)} (Ω) → ℝ,

I_{λ}(u) = ∫_{Ω} \frac{|∇u|^{p(x)}}{p(x)} dx − λ ∫_{Ω} \frac{a(x)}{q(x)} |u|^{q(x)} dx + λ ∫_{Ω} \frac{b(x)}{h(x)} |u|^{h(x)} dx.

We infer that for any x ∈ Ω and u ∈ W_{0}^{1,p(x)} (Ω)

\frac{λa(x)}{q(x)} |u(x)|^{q(x)} − \frac{λb(x)}{h(x)} |u(x)|^{h(x)}

≤ \frac{λa_{2}}{q_{1}} |u(x)|^{q(x)} − \frac{λb_{1}}{h_{2}} |u(x)|^{h(x)}

≤ \frac{λa_{2}}{q_{1}} (\frac{a_{2}h_{2}}{b_{1}q_{1}} )^{\frac{q(x)}{h(x)−q(x)}}

≤ \frac{λa_{2}}{q_{1}} [(\frac{a_{2}h_{2}}{b_{1}q_{1}} )^{\frac{q_{2}}{h_{1}−q_{2}}} + (\frac{a_{2}h_{2}}{b_{1}q_{1}} )^{\frac{q_{1}}{h_{2}−q_{1}}} ] := A  (3.17)

where A is a positive constant independent of u and x, by using the following elementary inequality [19, Lemma 4]

at^{k} − bt^{s} ≤ a (\frac{a}{b})^{\frac{k}{s−k}} , for all t ≥ 0

where a, b > 0 and 0 < k < s.

Integrating (3.17) over Ω, we get

λ ∫_{Ω} \frac{a(x)}{q(x)} |u(x)|^{q(x)} dx − λ ∫_{Ω} \frac{b(x)}{h(x)} |u(x)|^{h(x)} dx ≤ D

where D is a positive constant independent of u. Thus,

J_{λ}(u) ≥ \frac{1}{p_{2}} ∫_{Ω} |∇u|^{p(x)} dx − λ ∫_{Ω} \frac{a(x)}{q(x)} |u|^{q(x)} dx + λ ∫_{Ω} \frac{b(x)}{h(x)} |u|^{h(x)} dx

≥ \frac{1}{p_{2}} ∥u∥^{p_{1}} − D,

for all u ∈ W_{0}^{1,p(x)} (Ω) with ∥u∥ > 1. We infer that J_{λ}(u) → ∞ as ∥u∥ → ∞. Therefore the energy functional I_{λ} is coercive on W_{0}^{1,p(x)} (Ω). Moreover, a similar argument as the one used in the proof of [16, Lemma 3.4] shows that I_{λ} is also weakly lower semi-continuous in W_{0}^{1,p(x)} (Ω). These facts enable us to apply [22, Theorem 1.2] in order to find that there exits u_{λ} ∈ W_{0}^{1,p(x)} (Ω) a global minimizer of I_{λ} and thus, a weak solution of problem (P_{λ}^{/}).

Acknowledgments The authors would like to thank the referees and Professor Paul Eloe for their many helpful

suggestions and corrections, which improve the presentation of this paper.

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(Received January 27, 2010)