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

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