Qualitative Properties of Anisotropic Elliptic Schrödinger Equations

Abstract

Abstract We consider a class of nonlinear stationary Schrödinger-type equations and we are concerned with sufficient properties that guarantee the existence of multiple solutions in a suitable Sobolev space with variable exponents. We first establish that in the case of small perturbations, the problem admits at least two weak solutions. Next, in the case of convexconcave nonlinearities, we obtain conditions for the existence of infinitely many solutions with high (resp., small) energies. The arguments combine variational techniques with a careful analysis of the energy levels.