Extremal constant sign solutions and nodal solutions for the fractional p-Laplacian

TL;DR

This paper investigates the existence of extremal and nodal solutions for a fractional p-Laplacian equation, demonstrating the presence of extremal solutions within a specific order interval using variational and truncation methods.

Contribution

It introduces new results on the existence of extremal and nodal solutions for fractional p-Laplacian equations under Dirichlet conditions, employing a combination of variational and truncation techniques.

Findings

Existence of nonempty, directed, and compact solution set within the order interval.

Presence of extremal solutions: smallest positive and biggest negative solutions.

Existence of a nodal solution combining variational methods with truncation techniques.

Abstract

We study a pseudo-differential equation driven by the degenerate fractional p-Laplacian, under Dirichlet type conditions in a smooth domain. First we show that the solution set within the order interval given by a sub-supersolution pair is nonempty, directed, and compact, hence endowed with extremal elements. Then, we prove existence of a smallest positive, a biggest negative and a nodal solution, combining variational methods with truncation techniques.