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

We study a quasilinear elliptic problem $-\text{div}(\mathrm{\nabla }\mathrm{\Phi }(\mathrm{\nabla }u))+V(x)N^{\mathrm{\prime }}(u)=f(u)$ with anisotropic convex function $\mathrm{\Phi }$ 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 $W^1{L}^{\mathrm{\Phi }}({\mathbb{R}}^n)$. As the domain is unbounded we need to use Lions type lemma formulated for Young functions. Our assumptions broaden the class of considered functions $\mathrm{\Phi }$ so our result generalizes earlier analogous results proved in isotropic setting.