Quasilinear elliptic problem in anisotrpic Orlicz-Sobolev space on unbounded domain

Abstract

We study a quasilinear elliptic problem $-\text{div} (\nabla \Phi(\nabla u))+V(x)N'(u)=f(u)$ with anisotropic convex function $\Phi$ on whole $\mathbb{R}^n$. To prove existence of a nontrivial weak solution we use mountain pass theorem for a functional defined on anisotropic Orlicz-Sobolev space $W^1 L^{\Phi}(\mathbb{R}^n)$. As the domain is unbounded we need to use Lions type lemma formulated for Young functions. Our assumptions broaden the class of considered functions $\Phi$ so our result generalizes earlier analogous results proved in isotropic setting.