Five solutions for the fractional p-Laplacian with noncoercive energy

TL;DR

This paper establishes the existence of five solutions for a fractional p-Laplacian problem with a noncoercive energy functional, using critical point and Morse theory techniques under specific growth and resonance conditions.

Contribution

It introduces a novel approach to handle noncoercive fractional p-Laplacian problems with complex reaction behaviors, proving the existence of multiple solutions.

Findings

Existence of five nontrivial solutions including positive, negative, and nodal solutions.

Development of a method to analyze noncoercive energy functionals in fractional p-Laplacian problems.

Application of Morse theory and critical point theory to nonlinear fractional problems.

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.