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

Akrout Kamel

doi:10.1186/1687-2770-2011-50

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 ℝ n . This leads us to extend these results to m-equations systems. The method used is based on pohozaev type identities.

Highlights

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 [1,2,3,4,5,6,7,8]
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
In this work we are interested in the study of the nonexistence of nontrivial solutions for the p(x)-laplacian problem

Summary

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 [1,2,3,4,5,6,7,8]. In this work we are interested in the study of the nonexistence of nontrivial solutions for the p(x)-laplacian problem. Ω is bounded or unbounded domains of Rn, f is a locally lipshitzian function, H and p are given continuous real functions of C( ̄ ) verifying t. (., .) is the inner product in Rn We extend this technique to the system of m-equations. Where {fk} are locally lipshitzian functions verify fk(s1, ..., sk−1, 0, uk+1, ..., sm) = 0, (0 ≤ k ≤ m),. H is previously defined and pk functions of C1( ) class, verify pk(x) > 1, (x, ∇pk(x)) ≥ 0, ∀x ∈ ̄

Integral identities Let

Examples