Research

# Nonexistence of nontrivial solutions for the p(x)-Laplacian equations and systems in unbounded domains of ℝ n

Akrout Kamel

Author Affiliations

Department of mathematics and informatics. Tebessa university. Algeria

Boundary Value Problems 2011, 2011:50  doi:10.1186/1687-2770-2011-50

 Received: 12 January 2011 Accepted: 30 November 2011 Published: 30 November 2011

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

In this paper, we are interested on the study of the nonexistence of nontrivial solutions for the p(x)-Laplacian equations, in unbounded domains of ℝn. This leads us to extend these results to m-equations systems. The method used is based on pohozaev type identities.

### 1 Introduction

Several works have been reported by many authors, comprise results of nonexistence of nontrivial solutions of the semilinear elliptic equations and systems, under various situations, see [1-8]. The Pohozaĕv identity [1] published in 1965 for solutions of the Dirichlet problem proved absence of nontrivial solutions for some elliptic equations of the form

when Ω is a star shaped bounded open domain in ℝn and f is a continuous function on ℝ satisfying

A. Hareux and B. Khodja [2] established under the assumption

that the problems

admit only the null solution in H2(J × ω) ∩ L(J × ω). where J is an interval of ℝ and ω is a connected unbounded domain of ℝN such as

(n(x) is the outward normal to ∂ω at the point x)

In this work we are interested in the study of the nonexistence of nontrivial solutions for the p(x)-laplacian problem

(1.1)

with

where

Ω is bounded or unbounded domains of ℝn, f is a locally lipshitzian function, H and p are given continuous real functions of verifying

(1.4)

(., .) is the inner product in ℝn.

We extend this technique to the system of m-equations

(1.6)

with

Where {fk} are locally lipshitzian functions verify

H is previously defined and pk functions of class, verify

(1.9)

### 2 Integral identities

Let

with the norm

and

with the norm

Denote the closure of in W1, p(x) (Ω),

Lemma 1 Let solution of the equation (1.1) - (1.2), we have

(2.1)

Lemma 2 Let solution of the equation (1.1) - (1.3), we have

(2.2)

Proof Multiplying the equation (1.1) by and integrating the new equation by parts in Ω ∩ BR, BR = B (0, R)

Introducing the following result

we have

On the other hand

these results conduct to the following formula

(2.3)

Multiplying the present equation (1.1) by au and integrating the obtained equation by parts in Ω, we obtain

(2.4)

Combining (2.3) and (2.4) we obtain

On (Ω ∩ ∂BR) we have

so the last integral is major by

We remark that if Ω in bounded, so for R is little greater, we get Ω ∩ ∂BR = ϕ, then M (R) = 0.

If Ω is not bounded, such as |∇u| ∈ W1, p(x) (Ω), F(u) ∈ L1 (Ω) and we should see

consequently we can always find a sequence (Rn)n, such as

In the problem (1.1) - (1.2), u|Ω = 0. Then, , we obtain the identity (2.1).

In the problem (1.1) - (1.3), , we obtain the identity (2.2). ■

Lemma 3 Let , solution of the system (1.6) - (1.7). Then for the constants ak of ℝ, we have

(2.5)

Lemma 4 Let , solutions of the system (1.6) - (1.8). Then for the constants ak of ℝ, we have

(2.6)

Proof Multiplying the equation (1.6) by and integrating the new equation by part in Ω ∩ BR, BR = B (0, R), we get

On the other hand

These results conduct to the following formula

Doing the sum on k of 1 to m, we obtain

which leads to the following identity

(2.7)

Now, multiply the equation (1.1) by au and integrating the obtained equation by parts in Ω ∩ BR

(2.8)

Combining (2.7) and (2.8), we get the identities (2.5) and (2.6).

The rest of the proof is similar to the that of lemma 1. ■

### 3 Principal Result

theorem 3.1 If be a solution of the problem (1.1) - (1.2), Ω is star shaped and that a, H, f and F verify the following assumptions

(3.1)

(3.2)

Then, the problem admits only the null solution.

Proof Ω is star shaped, imply that

(3.3)

On the other hand, the condition (3.1) give

(3.4)

(1.4), (3.3) and (3.4), allow to get

So, the problem (1.1) - (1.2) becomes

(3.5)

Multiplying the equation (3.5) by u and integrating over Ω, we get

So

Hence u = cte = 0, because u|Ω = 0. ■

theorem 3.2 If solution of the problem (1.1) - (1.3), Ω is a star shaped and that a, H, F and F verify the following conditions

(3.6)

(3.7)

(3.8)

Therefore, the problem admits only the null solution.

Proof Similar to the proof of theorem 1. ■

theorem 3.3 If solution of the system (1.6) - (1.7), Ω is a star shaped and that ak, H, fk and Fm verify the following conditions

(3.9)

(3.10)

So, the system admits only the null solutions.

Proof Ω is a star shaped, implies that

(3.11)

On the other hand, the conditions (3.9) and (3.10), give

(3.12)

(1.4), (3.11) and (3.12), allow to have

So the system (1.6) - (1.7) becomes

(3.13)

Multiplying (3.13) by uk and integrating on Ω, we have

So

Therefore uk = cte = 0, ∀1 ≤ k m, because uk|Ω = 0. ■

theorem 3.4 If solution of the system (1.6) - (1.8), Ω is a star shaped and that ak, H, fk and Fm verify the following conditions

(3.14)

(3.15)

(3.16)

So, the problem admit only the null solution.

Proof Similar to the that of theorem 3. ■

### 4 Examples

Example 1 Considering in the following problem

(4.1)

where Ω is a bounded domain of n, c, μ > 0, q > 1 and

By choosing

we obtain

So, the problem (4.1) doesn't admit non trivial solutions if

Example 2 Considering in the following elliptic system

(4.2)

where Ω is a bounded domain of n, c, μ, γ, δ > 0 and p, q > 1.

By choosing

we obtain

So, the system (4.2) doesn't admit non trivial solutions if

### Competing interests

The author declares that they have no competing interests.

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