Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: An Orlicz–Sobolev space setting

We study the boundary value problem −div(log(1+q|∇u|)|∇u|p−2∇u)=f(u) in Ω, u=0 on ∂Ω, where Ω is a bounded domain in RN 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.