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

We study a quasilinear elliptic problem -div (∇ Φ (∇ u))+V(x)N'(u)=f(u) - div ( ∇ Φ ( ∇ u ) ) + V ( x ) N ′ ( u ) = f ( u ) with anisotropic convex function Φ Φ on the whole Rⁿ 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 \,W\,¹\,L\,^Φ (Rⁿ) W 1 L Φ ( 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 Φ Φ so our result generalizes earlier analogous results proved in isotropic setting.