The Nehari manifold approach for singular equations involving the p(<i>x</i>)-Laplace operator

Abstract

We study the following singular problem involving the p(x)-Laplace operator Δp(x)u=div(|∇u|p(x)−2∇u), where p(x) is a nonconstant continuous function, (Pλ){−Δp(x)u=a(x)|u|q(x)−2u(x)+λb(x)uδ(x)inΩ,u>0inΩ,u=0on∂Ω. Here, Ω is a bounded domain in RN≥2 with C2-boundary, λ is a positive parameter, a(x),b(x)∈C(Ω¯) are positive weight functions with compact support in Ω, and δ(x), p(x), q(x)∈C(Ω¯) satisfy certain hypotheses (A0) and (A1). We apply the Nehari manifold approach and some new techniques to establish the multiplicity of positive solutions for problem (Pλ).

Topics & Concepts