1. Introduction

In recent years, there has been an increasing interest in the study of various mathematical problems with variable exponents. These problems are interesting in applications (see [[1], [2]]). For the usual problems when p is constant, there are many results for existence of solutions when the domain is bounded or unbounded. For p variable, when the domain is bounded, on the results of existence of solutions, we refer to [[3], [4], [5]], when the domain is unbounded, results of existence of solutions are rare we can cite for example [[6], [7]].
In the case where is a bounded, and for 1 < p < N, In [8] authors studied the problem:

Under suitable smallness assumptions on f and g they prove the existence of a solution u which satisfies a further regularity. In [9] in the case of unbounded domains Guowei Dai By variational approach and the theory of the variable exponent Sobolev spaces establish the existence of infinitely many distinct homoclinic radially symmetric solutions whose W1;p(x)(RN)-norms tend to zero (to infinity, respectively) under weaker hypotheses about nonlinearity at zero (at infinity, respectively). The principal objective of this paper is to prove the existence and some regularity of solutions of the following p(x)-Laplacian equation in open set of RN (possibly of infinite measure):

where p is log-H¨older continuous function such that 1 < p?  p+ < N,
p(x)(u) = div(jrujp(x)?2ru) is the p(x)-Laplace operator, 0 is a positive constant, d is a function in L1(
). We assume the following hypotheses on the source terms f and g : f : ! R, g : ! RN are a measurable function

We will proceed by solving the problem on a sequence n of bounded sets after that we pass to the limit in the approximating problems by using the a prior estimate (this a prior estimates provide the necessary compactness properties for solutions) from which the desired results are easily inferred. To this aim, we can neither use any embedding theorem between Lp(:)( ) nor any argument involving the measure of n, and under suitable assumptions on f and g we prove some regularity of a solutions u of (1). A similar result has been proved in [7] where p is constant such that 1 < p < N but in the present setting such an approach cannot be used directly, because of the variability of p.  The plan of the paper is the following: In Section 2 we recall some important defnitions and results of variable exponent Lebesgue and Sobolev spaces. In Section 3 we will give the precise assumptions and
state the main results. In Section 4 we will define the approximate problems, state the a priori estimates that we want to obtain. In the Sections which follow we will prove strong convergence of un and their gradients run. Section 5 is devoted to conclude the proof of the main existence results. Finally, in Section 6, we prove that, if f and g have higher integrability, then every solution u of (1) is bounded. More precisely, we will assume that (2) are replaced by:

2. Preliminaries

In order to discuss the problem (1), we need to recall some definitions and basic properties of Lebesgue and Sobolev spaces with variable exponents. Let an open bounded set of RN with N  2. We say that a real-valued continuous function p(:) is log- H¨older continuous in if:

For all p1;p2 2 C+ such that: p1(x)  p2(x) a.e. in , we have: Lp2(x)  ,!Lp1(x) and the embedding is continuous.
Proposition 2 ([10, 11]) If we denote

(ii) If q 2 C+ and q(x) < p(x) for any x 2, then the embedding W1;p(x)0  ,!,!Lq(x) is continuous and compact.
(iii) Poincar´e inequality: There exists a constant C > 0,such that:kuk Lp(x)  Ckruk Lp(x) 8u 2W1;p(x)0 :
(vi) Sobolev-Poincar´e inequality : there exists an other constant C > 0, such that:kuk Lp(x)  Ckruk Lp(x) 8u 2W1;p(x)0 : The symbol * will denote the weak convergence, and the constants Ci , i = 1;2; : : : used in each step of proof are independent.

3. Approximate problems and A priori estimates

In this section we will prove the existence result to the approximate problems. Also we will give a uniform estimate for this solutions un. Approximate problems For k > 0 and s 2 R, the truncation function Tk(:) is defined by:

Lemma 3 The operator Bn = A+Rn is pseudo-monotone from W 1;p(x) 0 (n) into W?1;p0 (x)(n). Moreover, Bn is coercivein the following sense

A priori estimates

Proposition 4 Assuming that p(:) 2 C+ holds, and let un be any solution of (5). Then for every  > 0 there exists a positive constant C = C(N;p;0;d; f ;g;) such that:

Now let us observe that p is a continuous variable exponent on then there exists a constant  > 0 such that:

Putting all the inequalities (22), (25), (32), (33), (35), (38), (41), (37) and (36) together, we get an estimate in W 1;p(x) 0 for Gk(u),when k is large enough:

Remark 3 if meas is finite or if f 2 L1 it is easy to estimate the integral L3 In general case, let  be a positive constant to bechosen later, we write

For every  such that (23) , where C17 depends on  and on the data of the problem. Note that (49) does not imply an estimate in Lp(x) for ejuj?1, since meas may be infinite. To obtain such an estimate, we have to combine (48) and (49) , since, for every k > 0, one has the inequalities

4 Main results

In this section we will prove the main result of this paper. Let fung be any sequence of solutions of problem (5), we extend them to zero in n n. By (20), there exist a subsequence (still denoted by un) and a function u 2W 1;p(x) 0 ( ) such that un *u weakly in W 1;p(x) 0 ( ). Theorem 1 There exists at least one solution u of (1); which is such that

If  is an arbitrary positive number, let us choose H such that the right-hand side of (53) is smaller than . It follows that, for every k satisfying (34), (40) , every  satisfying (23), and every n 2N

Splitting into = fjunj  kg[fjunj > kg we can write

which is a fixed function in Lp0 (x)(). Therefore by Lebesgue’s theorem we have

Let us examine Cn and Dn together. We first fix  such that  > d Since ‘(zn)sign(un) = j'(zn)j on the set fjunj > kg we have

For the term Fn we can see that jr jj’znj converge strongly to zero in Lr(x)( ) for every r(x) > 1. by (20) the term jrunjp(x)?2runejunj is bounded in Lp0 (x) loc ( ) then we have that:

To obtain (51) we have to pass to the limit in the distributional formulation of problem (5) using (69). Finally, statement (52) follows easily from Proposition 4 and (69), using Fatou’s Lemma.

5. Boundedness of solutions

In this section we will gave some regularity on the solution of the problem (1) using an adaptation of a classical technique due to Stampacchia. To do this we need the following lemma (see [15]):  Lemma 4 Let  be a non-negative, non-increasing function defined on the halfline [k0;1). Suppose that there exist positive constants A, ,, with  > 1, such that

The proof relies on the combined use of the wellknown technique by Stampacchia (see [15]) and suitable exponential test functions, as in [16].

the second integral in the right-hand side of (74) can be absorbed by the left-hand side. In view of H¨older’s inequality and (71) and (3) we have:

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