Existence and multiplicity results for anisotropic stationary Schrödinger equations

Abstract

In this paper we are concerned with a new class of quasilinear elliptic equations with a power-like reaction term and a di¤erential operator that involves partial derivatives with different powers. The functional-analytic framework relies on anisotropic Sobolev spaces. By means of combined variational arguments, we obtain the existence of weak solutions and, in case of symmetric settings, the existence of large or small energy solutions. In particular, we establish some results that extend the classical theory of combined e¤ects of concave and convex nonlinearities.