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We establish some results on the existence of multiple nontrivial solutions for a class of p(x)-Lap-lacian elliptic equations without assumptions that the domain is bounded. The main tools used in the proof are the variable exponent theory of generalized Lebesgue-Sobolev spaces, variational methods and a variant of the Mountain Pass Lemma.

The study of differential and partial differential equations involving variable exponent conditions is a new and interesting topic. The interest in studying such problem was stimulated by their applications in elastic mechanics and fluid dynamics. These physical problems were facilitated by the development of Lebesgue and Sobolev spaces with variable exponent.

The existence and multiplicity of solutions of

In [

Also Xiaoyan Lin and X. H. Tang in [

and they proved the multiplicity of solutions for problem (2) by using the cohomological linking method for cones and a new direct sum decomposition of

In this paper, we consider the following problem

where

(B_{0})

here m is the Lebesgue measure on R^{N}.

(F_{0})

for some

Define the subspace

and the functional

where

Clearly, in order to determine the weak solutions of problem (3), we need to find the critical points of functional Φ. It is well known that under (B_{0}) and (F_{0}), Φ is well defined and is a C^{1} functional. Moreover,

for all

If

Set

This paper is to show the existence of nontrivial solutions of problem (3) under the following conditions.

(F_{1})

where

(F_{2}) There exist

(F_{3}) There exist

for a.e.

(F_{4}) _{0}).

We have the following results.

Theorem 1.1. If _{0}), _{0}), (F_{1}) and (F_{2}), then problem (3) possesses at least one nontrivial solution.

Theorem 1.2. Assume _{0}), _{0}), (F_{3}) and (F_{4}), with

This paper is divided into three sections. In the second section, we state some basic preliminary results and give some lemmas which will be used to prove the main results. The proofs of Theorem 1.1 and Theorem 1.2 are presented in the third section.

In this section, we recall some results on variable exponent Sobolev space

Let

For

Define the variable exponent Sobolev space:

which is endowed with the norm

It can be proved that the spaces

Proposition 2.1. [

Then we have

1) For

2)

3)

4)

For

We have the following generalized Hölder type inequality.

Proposition 2.2. [

We consider the case that _{0}). Define the norm

Then

Similar to the Proposition 2.1, we have

Proposition 2.3. [

has the following properties:

1)

2)

3)

Lemma 2.4. [_{0}), then

1) we have a compact embedding

2) for any measurable function

Now, we consider the eigenvalues of the p(x)-Laplacian problem

For any

For all

then

where

Define

We denote by

Define

Lemma 2.5. For all

Proof. Let

On the other hand, since the functional

Lemma 2.6.

Proof. From Lemma 2.5, we have

Since

so we have

Thus we get

Similarly, if

On the other hand, it is easy to see that

Now, we consider the truncated problem

where

We denote by

Lemma 2.7.

1) If

2) The mappings

Lemma 2.8. All solutions of

Proof. Define

where _{0}), ^{1}-functionals.

Let u be a solution of

we have

By virtue of Proposition 2.3, we have

Similarly, the nontrivial critical points of the functional

To derive the Theorem 1.1, we need the following results.

Proposition 3.1. Φ is coercive on E.

Proof. Put

From (F_{1}) we have, for any

By contradiction, let

Putting

Consequently,

Set

then,

From (6), (F_{1}) and Lemma 2.6, we deduce that

This is a contradiction. Therefore, Φ is coercive on E. ,

Proposition 3.2. Assume _{0}) and (F_{2}), then zero is local maximum for the functional

Proof. From (F_{2}), there is a constant

From (F_{0}) and

By (7) and (8), we get

For

Since

Thus the proposition follows. ,

Proof of Theorem 1.1. From Proposition 3.1, we know Φ is coercive on E. Since Φ has a global minimizer

To find the properties of the p(x)-Laplacian operators, we need the following inequalities (see [

Lemma 3.3. For ^{N}, then there are the following inequalities

Proposition 3.4. Assume (F_{0}), and let ^{N}, as

Proof. Let

Let us denote by

From Lemma 3.3, we have

Recalling that

and so,

On the other hand, by (11) and

Thus,

Combining Hölder’s inequality and Sobolev embedding, we deduce that

Let us consider the sets

From Lemma 3.3, we get

Applying again Hölder’s inequality,

where

and

Then,

From (12) and (13), we have

By (15)-(17), we obtain

(12) and (14) imply that

From (18) and (19),

Combined with Lebesgue’s dominated convergence theorem, we get

By (20) and

Proposition 3.5. Assume (F_{0}), and let _{d} sequence in E for

Proof. From (F_{0}), we have

Assume that

Since

Proposition 3.6. Assume _{0}), _{0}) and (F_{4}), and let _{d} sequence in E, then

Proof. From Proposition 3.4, we have

By Lemma 2.4, we get

On the other hand, Lebesgue’s dominated convergence theorem and the weak convergence of

Moreover, since

Therefore, by virtue of the definition of weak convergence, we obtain

By (23)-(25), we have

By (22) and (26), we get

Then combining Lemma 3.3, we obtain

which imply that

Proposition 3.7. There exist

Proof. The conditions (F_{0}) and (F_{4}) imply that

For

By the condition (F_{0}), it follows

from Lemma 2.4, which implies the existence of

Using (28) and Proposition 2.1, we deduce

Combining (27), it results in that

here

Since

,

Proposition 3.8. If

Proof. From the condition (F_{3}), we get

For

Since

,

Proof of Theorem 1.2. According to the Mountain Pass Lemma, the functional

Similarly, for functional

This work has been supported by the Natural Science Foundation of China (No. 11171220) and Shanghai Leading Academic Discipline Project (XTKX2012) and Hujiang Foundation of China (B14005).

HonghongQi,GaoJia, (2015) Existence and Multiplicity of Solutions for Quasilinear p(x)-Laplacian Equations in R^{N}. Journal of Applied Mathematics and Physics,03,1270-1281. doi: 10.4236/jamp.2015.310156