Five solutions for the fractional p-Laplacian with noncoercive energy

Abstract

We deal with a Dirichlet problem driven by the degenerate fractional p-Laplacian and involving a nonlinear reaction which satisfies, among other hypotheses, a (p-1)-linear growth at infinity with non-resonance above the first eigenvalue. The energy functional governing the problem is thus noncoercive. Thus we focus on the behavior of the reaction near the origin, assuming that it has a (p-1)-sublinear growth at zero, vanishes at three points, and satisfies a reverse Ambrosetti-Rabinowitz condition. Under such assumptions, by means of critical point theory and Morse theory, and using suitably truncated reactions, we show the existence of five nontrivial solutions: two positive, two negative, and one nodal.