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

Abstract

AbstractWe study a quasilinear elliptic problem $$-\text {div} (\nabla \Phi (\nabla u))+V(x)N'(u)=f(u)$$ - div ( ∇ Φ ( ∇ u ) ) + V ( x ) N ′ ( u ) = f ( u ) with anisotropic convex function $$\Phi $$ Φ on the whole $$\mathbb {R}^n$$ 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 $${{{\,\mathrm{\textbf{W}}\,}}^1}{{\,\mathrm{\textbf{L}}\,}}^{{\Phi }} (\mathbb {R}^n)$$ 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 $$\Phi $$ Φ so our result generalizes earlier analogous results proved in isotropic setting.