Uniqueness of least energy solutions to the fractional Lane-Emden equation in the ball

TL;DR

This paper proves the uniqueness of least-energy solutions to the fractional Lane-Emden equation in a ball, using Morse theory and a new Hopf's Lemma-type result, advancing understanding of fractional PDEs.

Contribution

It establishes the uniqueness of least-energy solutions for the fractional Lane-Emden equation in a ball, introducing a new Hopf's Lemma-type result and applying Morse theory.

Findings

Uniqueness of least-energy solutions in a ball for the fractional Lane-Emden equation.

Development of a new Hopf's Lemma-type result for fractional PDEs.

Application of Morse theory to prove solution nondegeneracy.

Abstract

We prove uniqueness of least-energy solutions to the fractional Lane-Emden equation, under homogeneous Dirichlet exterior conditions, when the underlying domain is a ball $B \subset \mathbb{R}^N$. The equation is characterized by a superlinear, subcritical power-like nonlinearity. The proof makes use of Morse theory and is inspired by some results obtained by C. S. Lin in the '90s. A new Hopf's Lemma-type result shown in this paper is an essential element in the proof of nondegeneracy of least-energy solutions.