Five solutions for the fractional p-Laplacian with noncoercive energy

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 nonresonance above the first eigenvalue. The energy functional governing the problem is thus noncoercive. 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.

Automated summary

This study investigates a Dirichlet problem driven by the degenerate fractional p-Laplacian and involving a nonlinear reaction. The research shows that under certain conditions, there are five nontrivial solutions for the problem, including two positive solutions, two negative solutions, and one nodal solution.