Quasilinear elliptic problem in anisotropic Orlicz–Sobolev space on unbounded domain

We study a quasilinear elliptic problem \(-\text {div} (\nabla \Phi (\nabla u))+V(x)N'(u)=f(u)\) with anisotropic convex function \(\Phi \) on the whole \(\mathbb {R}^n\). To prove existence of a nontrivial weak solution we use the mountain pass theorem for a functional defined on anisotropic Orlicz–Sobolev space \({{{\,\mathrm{\textbf{W}}\,}}^1}{{\,\mathrm{\textbf{L}}\,}}^{{\Phi }} (\mathbb {R}^n)\). 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.