Coercive and noncoercive nonlinear Neumann problems with indefinite potential

Abstract

Abstract We consider nonlinear Neumann problems driven by a nonhomogeneous differential operator and an indefinite potential. In this paper we are concerned with two distinct cases. We first consider the case where the reaction is (p-1)-sublinear near ±∞ and (p-1)-superlinear near zero. In this setting the energy functional of the problem is coercive. In the second case, the reaction is (p-1)-superlinear near ±∞ (without satisfying the Ambrosetti–Rabinowitz condition) and has a (p-1)-sublinear growth near zero. Now, the energy functional is indefinite. For both cases we prove “three solutions theorems” and in the coercive setting we provide sign information for all of them. Our approach combines variational methods, truncation and perturbation techniques, and Morse theory (critical groups).