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

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