Multiplicity of nonnegative solutions for semilinear Robin problems involving sign‐changing nonlinearities

Abstract In this article, we investigate the existence and multiplicity of solutions to the Robin problem where () is a smooth bounded domain, and . Our main assumption is that is a locally Lipschitz function, possibly sign‐changing, such that for every , where are two zeros of . Without any further conditions, we establish the existence of two nonnegative solutions whose maximum lies in for sufficiently large . Moreover, we analyse the limiting behaviour of the solution set of this Robin problem, showing that it degenerates into that of the associated Neumann problem as and into that of the associated Dirichlet problem as .