Second radial eigenfunctions to a fractional Dirichlet problem and uniqueness for a semilinear equation

TL;DR

This paper studies the properties of second radial eigenfunctions of fractional Schrödinger operators in a ball, proving their shape, boundary behavior, and applying these results to establish uniqueness of ground state solutions for a nonlinear fractional PDE.

Contribution

It introduces a new analysis of second radial eigenfunctions for fractional operators and proves a novel boundary derivative property, leading to uniqueness results for nonlinear fractional equations.

Findings

Second radial eigenfunctions change sign exactly once.

Eigenfunctions have nonvanishing fractional boundary derivatives.

Uniqueness and nondegeneracy of ground state solutions are established.

Abstract

We analyze the shape of radial second Dirichlet eigenfunctions of fractional Schr\"odinger type operators of the form $(-\Delta)^s +V$ in the unit ball $B$ in $\mathbb{R}^N$ with a nondecreasing radial potential $V$. Specifically, we show that the eigenspace corresponding to the second radial eigenvalue is simple and spanned by an eigenfunction $u$ which changes sign precisely once in the radial variable and does not have zeroes anywhere else in $B$. Moreover, by a new Hopf type lemma for supersolutions to a class of degenerate mixed boundary value problems, we show that $u$ has a nonvanishing fractional boundary derivative on $\partial B$. We apply this result to prove uniqueness and nondegeneracy of positive ground state solutions to the problem $(-\Delta)^s u+\lambda u=u^p$ on ${B}$, $\; u=0$ on $\mathbb{R}^N\setminus B$. Here $s\in (0,1)$, $\lambda\geq 0$ and $p>1$ is strictly smaller than the critical Sobolev exponent.