Four solutions for fractional p-Laplacian equations with asymmetric reactions

TL;DR

This paper studies a nonlinear fractional p-Laplacian equation with asymmetric reactions, proving the existence of at least four solutions for small parameters using critical point and Morse theories.

Contribution

It introduces a new approach combining critical point theory, Morse theory, and a Brezis-Oswald comparison to establish multiple solutions for asymmetric fractional p-Laplacian problems.

Findings

Existence of at least four solutions for small parameters

Identification of positive, negative, and nodal solutions

Development of a Brezis-Oswald type comparison result

Abstract

We consider a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degenerate fractional p-Laplacian, whose reaction combines a sublinear term depending on a positive parameter and an asymmetric perturbation (superlinear at positive infinity, at most linear at negative infinity). By means of critical point theory and Morse theory, we prove that, for small enough values of the parameter, such problem admits at least four nontrivial solutions: two positive, one negative, and one nodal. As a tool, we prove a Brezis-Oswald type comparison result.