Stability of solutions for nonlocal problems

TL;DR

This paper investigates the stability and convergence of solutions, eigenvalues, and ground states of fractional p-Laplace problems as the fractional parameter approaches 1, unifying various stability results.

Contribution

It provides a general convergence theorem for weak solutions of fractional p-Laplace problems as s approaches 1, including applications to eigenvalues and ground states.

Findings

Weak solutions converge as s approaches 1

Ground state solutions exhibit stability in the limit

p-fractional eigenvalues are stable as s approaches 1

Abstract

In this paper we deal with the stability of solutions of fractional $p-$Laplace problems with nonlinear sources when the fractional parameter $s$ goes to 1. We prove a general convergence result for general weak solutions which is applied to study the convergence of ground state solutions of $p-$fractional problems in bounded and unbounded domains as $s$ goes to 1. Moreover, our result applies to treat the stability of $p-$fractional eigenvalues as $s$ goes to 1.