Multiple solutions for superlinear Dirichlet problems with an indefinite potential

Abstract

We consider a semilinear elliptic equation with an indefinite unbounded potential and a Carathéodory reaction term that exhibits superlinear growth near ±∞ without satisfying the AR-condition. Also, at the origin, the primitive of the reaction satisfies a nonuniform nonresonance condition with respect to the first eigenvalue of $${{\left(-\Delta,H_0^1(\Omega)\right)}}$$ . Using critical point theory and Morse theory, we show that the problem has at least three nontrivial smooth solutions. Our result extends that of Wang (Anal Nonlineaire 8:43–58, 1991).