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

TL;DR

This paper establishes the existence of nontrivial solutions for a class of quasilinear elliptic problems in anisotropic Orlicz-Sobolev spaces on unbounded domains, extending previous isotropic results using variational methods.

Contribution

It broadens the class of admissible functions  specifically anisotropic convex functions  for quasilinear elliptic problems, generalizing earlier isotropic case results.

Findings

Existence of nontrivial weak solutions proved.

Use of mountain pass theorem in anisotropic Orlicz-Sobolev spaces.

Extension of Lions lemma for Young functions on unbounded domains.

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.