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

We study the boundary value problem −div(log(1 + |∇u|q )|∇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 +q 0. 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. © 2006 Elsevier Inc. All rights reserved