Non-uniformly elliptic equations with non-uniformly $p$-superlinear nonlinearities

Abstract

The purpose of this paper is to study the non-uniformly elliptic equations with non-uniformly $p$-superlinear nonlinearities satisfying sets of other nonuniform conditions. In particular, we weaken the well-known Ambrosetti-Rabinowitz condition. By introducing the notion of pseudo-uniform convexity, we complete the studies in [5], [10] for non-uniformly p-Laplacian problems with $1 < p < 2$. We are able to prove the existence of nontrivial, non-negative, non-positive solutions.