Existence and Multiplicity of Solutions for Quasilinear p(x)-Laplacian Equations in R^N

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.

• p(x)-Laplacian Operator
• Generalized Lebesgue-Sobolev Spaces
• Variational Method
• Multiple Solutions

1. Introduction

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 -Laplacian problems have been studied by several authors (see for example [1] [2] , and the references therein).

In [3] , A. R. EL Amrouss and F. Kissi proved the existence of multiple solutions of the following problem

Also Xiaoyan Lin and X. H. Tang in [4] studied the following quasilinear elliptic equation

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 is the -Laplacian operator; is a Lipschitz continuous function with

is a given continuous function which satisfies

(B₀)

here m is the Lebesgue measure on R^N.

is a Carathéodory function satisfying the subcritical growth condition

(F₀)

for some, where, , , and

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 (B₀) and (F₀), Φ is well defined and is a C¹ functional. Moreover,

for all.

If for a.e., the constant function is a trivial solution of problem (3). In the following, the key point is to prove the existence of nontrivial solutions for problem (3).

Set

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

(F₁)

where as given in (4).

(F₂) There exist and, such that

(F₃) There exist and such that

for a.e..

(F₄) as and uniformly for, with. Here is given in the condition (F₀).

We have the following results.

Theorem 1.1. If satisfies (B₀), satisfies (F₀), (F₁) and (F₂), then problem (3) possesses at least one nontrivial solution.

Theorem 1.2. Assume satisfies (B₀), satisfies (F₀), (F₃) and (F₄), with for a.e., then problem (3) has at least two nontrivial solutions, in which one is non-negative and another is non-positive.

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.

2. Preliminaries

In this section, we recall some results on variable exponent Sobolev space and basic properties of the variable exponent Lebesgue space, we refer to [5] -[8] .

Let,. Define the variable exponent Lebesgue space:

For, we define the following norm

Define the variable exponent Sobolev space:

which is endowed with the norm

It can be proved that the spaces and are separable and reflexive Banach spaces. See [9] for the details.

Proposition 2.1. [10] [11] Let

Then we have

1) For,;

2),

3)

4),

For with, let satisfy

We have the following generalized Hölder type inequality.

Proposition 2.2. [9] [12] For any and, we have

We consider the case that satisfies (B₀). Define the norm

Then is continuously embedding into as a closed subspace. Therefore, is also a separable and reflexive Banach space.

Similar to the Proposition 2.1, we have

Proposition 2.3. [13] The functional defined by

has the following properties:

1);

2)

3)

Lemma 2.4. [13] If satisfies (B₀), then

1) we have a compact embedding;

2) for any measurable function with, we have a compact embedding

. Here means that.

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

For any, define by

For all, set

then is a submanifold of E since t is a regular value of H. Put

where is the genus of I.

Define

We denote by the eigenpair sequences of problem (5) such that

Define

, where

Lemma 2.5. For all, let be an eigenfunction associated with of the problem (5). Then,

Proof. Let. From the definition of, it is easy to see that.

On the other hand, since the functional is coercive and weakly lower semi-continuous and is weakly closed subset of E, there exists such that. Letting, then and. Thus the lemma follows. ,

Lemma 2.6.

Proof. From Lemma 2.5, we have

Since

so we have Then,

Thus we get and.

Similarly, if is the eigenfunction associated with, we get and. Finally, we obtain

On the other hand, it is easy to see that Thus the lemma follows. ,

Now, we consider the truncated problem

where

We denote by and the positive and negative parts of u.

Lemma 2.7.

1) If then and

2) The mappings are continuous on E.

Lemma 2.8. All solutions of (resp.) are non-positive (resp. non-negative) solutions of problem (3).

Proof. Define

where From Lemma 2.7 and (F₀), is well defined on E, weakly lower semi-con- tinuous and C¹-functionals.

Let u be a solution of. Taking in

we have

By virtue of Proposition 2.3, we have, so and, a.e., then u is also a critical point of the functional Φ with critical value.

Similarly, the nontrivial critical points of the functional are non-negative solutions of problem. ,

3. Proof of Main Results

3.1. Proof of Theorem 1.1

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

Proposition 3.1. Φ is coercive on E.

Proof. Put

From (F₁) we have, for any, there is such that

By contradiction, let and such that

Putting, one has. For a subsequence, we may assume that for some, we have

weakly in E and strongly in.

Consequently,. Let, via the result above we have and

Set

then,

From (6), (F₁) and Lemma 2.6, we deduce that

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

Proposition 3.2. Assume satisfies (F₀) and (F₂), then zero is local maximum for the functional, ,.

Proof. From (F₂), there is a constant such that

From (F₀) and, there exists such that

By (7) and (8), we get

For we have

Since, there is a such that

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 on E, Φ is weakly lower semi-continuous and, then, in order to prove, we need to prove. So we have the Theorem 1.1 following from Proposition 3.2. ,

3.2. Proof of Theorem 1.2

To find the properties of the p(x)-Laplacian operators, we need the following inequalities (see [10] ).

Lemma 3.3. For and in R^N, then there are the following inequalities

Proposition 3.4. Assume (F₀), and let be a sequence such that in E and for all as, then, for some subsequences, , a.e. in R^N, as and for all.

Proof. Let and such that

Let us denote by the following sequence

From Lemma 3.3, we have and

Recalling that in E, we get

and so,

On the other hand, by (11) and we obtain

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), a.e. in. Because R is arbitrary, it follows that for some subsequence

.

Combined with Lebesgue’s dominated convergence theorem, we get

By (20) and, we derive that for all. ,

Proposition 3.5. Assume (F₀), and let and be a (PS)_d sequence in E for then is bounded in E.

Proof. From (F₀), we have It is clear that

Assume that for some, then, by Proposition 2.3, Hölder’s inequality and Sobolev embedding, we have

Since and, (21) implies that is bounded in E. ,

Proposition 3.6. Assume satisfies (B₀), satisfies (F₀) and (F₄), and let be a (PS)_d sequence in E, then satisfies the (PS) condition.

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 to u in E show

Moreover, since are bounded in, then we have

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 in E. ,

Proposition 3.7. There exist and such that, for all with.

Proof. The conditions (F₀) and (F₄) imply that

For small enough, combined with Proposition 2.3, we have

By the condition (F₀), it follows

from Lemma 2.4, which implies the existence of such that

Using (28) and Proposition 2.1, we deduce

Combining (27), it results in that

here are positives constants. Taking such that we obtain

Since, the function is strictly positive in a neighborhood of zero. It follows that there exist and such that

,

Proposition 3.8. If and, we have for a certain.

Proof. From the condition (F₃), we get

For and, we have

Since, we obtain

,

Proof of Theorem 1.2. According to the Mountain Pass Lemma, the functional has a critical point with. But, , that is, , a.e.. Therefore, the problem has a nontrivial solution which, by Lemma 2.8, is a non-positive solution of the problem (3).

Similarly, for functional, we still can show that there exists furthermore a non-negative solution. The proof of Theorem 1.2 is now complete. ,

Acknowledgements

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).

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