On a class of elliptic problems with indefinite nonlinearities

Abstract

In this paper we apply variational and sub-supersolution methods to study the existence and multiplicity of nonnegative solutions for a class of indefinite semilinear elliptic problems that depend on a parameter. The results on the existence of solutions do not impose any growth condition at infinity on the term which depends on the parameter. To derive such results, first we find a positive supersolution by solving an auxiliary problem. Then we use a truncation argument and a global minimization method. The main hypothesis for the existence of two nonzero solutions is that the indefinite term is the product of a weight function, having a thick zero set, and a nonlinear function which satisfies the Ambrosetti–Rabinowitz superlinear condition. Results for some corresponding indefinite problems are also established.