Compactness and existence results for quasilinear elliptic problems with singular or vanishing potentials

Abstract

Given [Formula: see text], [Formula: see text], two measurable functions [Formula: see text] and [Formula: see text], and a continuous function [Formula: see text] ([Formula: see text]), we study the quasilinear elliptic equation [Formula: see text] We find existence of nonnegative solutions by the application of variational methods, for which we have to study the compactness of the embedding of a suitable function space [Formula: see text] into the sum of Lebesgue spaces [Formula: see text], and thus into [Formula: see text] ([Formula: see text]) as a particular case. Our results do not require any compatibility between how the potentials [Formula: see text], [Formula: see text] and [Formula: see text] behave at the origin and at infinity, and essentially rely on power type estimates of the relative growth of [Formula: see text] and [Formula: see text], not of the potentials separately. The nonlinearity [Formula: see text] has a double-power behavior, whose standard example is [Formula: see text], recovering the usual case of a single-power behavior when [Formula: see text].