Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: an Orlicz-Sobolev space setting

2006/06/07 by Mihailescu, Mihai, Radulescu, Vicentiu
#35D05 #35J60 #35J70 #46N20 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.math/0606157

Abstract

We study the boundary value problem -\rm div(log(1+ |∇ u|^q)|∇ u|^p-2∇ u)=f(u) in Ω, u=0 on ∂Ω, where Ω is a bounded domain in \RR^N with smooth boundary. We distinguish the cases where either f(u)=-λ|u|^p-2u+|u|^r-2u or f(u)=λ|u|^p-2u-|u|^r-2u, with p, q>1, p+q0. In the second case we prove the existence of a nontrivial weak solution if λ is sufficiently large. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces.

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