Boundary Blow-Up Solutions to p(x)-Laplacian Equations with Exponential Nonlinearities

Qihu Zhang

doi:10.1155/2008/279306

Abstract

This paper investigates the -Laplacian equations with exponential nonlinearities in , as , where is called - Laplacian. The singularity of boundary blow-up solutions is discussed, and the existence of boundary blow-up solutions is given.

Highlights

The study of differential equations and variational problems with nonstandard p x -growth conditions is a new and interesting topic
Many results have been obtained on this kind of problems, for example, 1–15
Throughout the paper, we assume that p x and f x, u satisfy that

Summary

Introduction

The study of differential equations and variational problems with nonstandard p x -growth conditions is a new and interesting topic. We consider the p x -Laplacian equations with exponential nonlinearities. −Δp x u ef x,u 0 in Ω, P u x −→ ∞ as d x, ∂Ω −→ 0, where −Δp x u −div |∇u|p x −2∇u , Ω B 0, R ⊂ RN is a bounded radial domain B 0, R {x ∈ RN | |x| < R}. The operator −Δp x u −div |∇u|p x −2∇u is called p x -Laplacian. If p x ≡ p a constant , P is the well-known p-Laplacian problem see 16–18. Because of the nonhomogeneity of p x -Laplacian, p x -Laplacian problems are more complicated than those of p-Laplacian ones see 6 ; and another difficulty of this paper is that f x, u cannot be represented as h x f u

Preliminary

Main results and proofs