Elliptic equations with indefinite and unbounded potential and a nonlinear concave boundary condition

Abstract

We consider an elliptic problem driven by the negative Laplacian plus an indefinite and unbounded potential and a superlinear reaction. The boundary condition is parametric, nonlinear and superlinear near zero. Thus, the problem is a new version of the classical “convex–concave” problem (problem with competing nonlinearities). First, we prove a bifurcation-type result describing the set of positive solutions as the parameter [Formula: see text] varies. We also show the existence of a smallest positive solution [Formula: see text] and investigate the properties of the map [Formula: see text]. Finally, by imposing bilateral conditions on the reaction we generate two more solutions, one of which is nodal.