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

Mouhamed Moustapha Fall, Tobias Weth

TL;DR

The paper analyzes the radial second eigenfunctions of the fractional Dirichlet operator $(-\Delta)^s+V$ on the unit ball with radial, nondecreasing $V$, proving that the second radial eigenvalue $\sigma_2(V)$ is simple and its eigenfunction $w_2$ changes sign exactly once in the radial variable; it also establishes a nonvanishing fractional boundary derivative via a new Hopf-type lemma. Leveraging a rearrangement argument and a continuation from the classical Dirichlet Laplacian, the authors obtain a sharp nodal structure, including a two-nodal-domain property for the $s$-harmonic extension, and derive a local Hopf boundary behavior $\psi_{w_2}(1)<0$ under regularity of $V$. These spectral results feed into a nonlinear analysis showing uniqueness and nondegeneracy of positive ground state solutions to the semilinear problem $(-\Delta)^s u+\lambda u=u^p$ in $B$ with $u=0$ outside, for subcritical $p$; the approach relies on a combination of rearrangement, variational characterizations, and a localized fractional Hopf lemma. Overall, the work extends understanding of nodal properties and boundary behavior for nonlocal Dirichlet problems and their nonlinear consequences in a bounded domain.

Abstract

We analyze the shape of radial second Dirichlet eigenfunctions of fractional Schrödinger type operators of the form $(-Δ)^s +V$ 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 $(-Δ)^s u+λu=u^p$ on ${B}$, $\; u=0$ on $\mathbb{R}^N\setminus B$. Here $s\in (0,1)$, $λ\geq 0$ and $p>1$ is strictly smaller than the critical Sobolev exponent.

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

TL;DR

The paper analyzes the radial second eigenfunctions of the fractional Dirichlet operator on the unit ball with radial, nondecreasing , proving that the second radial eigenvalue is simple and its eigenfunction changes sign exactly once in the radial variable; it also establishes a nonvanishing fractional boundary derivative via a new Hopf-type lemma. Leveraging a rearrangement argument and a continuation from the classical Dirichlet Laplacian, the authors obtain a sharp nodal structure, including a two-nodal-domain property for the -harmonic extension, and derive a local Hopf boundary behavior under regularity of . These spectral results feed into a nonlinear analysis showing uniqueness and nondegeneracy of positive ground state solutions to the semilinear problem in with outside, for subcritical ; the approach relies on a combination of rearrangement, variational characterizations, and a localized fractional Hopf lemma. Overall, the work extends understanding of nodal properties and boundary behavior for nonlocal Dirichlet problems and their nonlinear consequences in a bounded domain.

Abstract

We analyze the shape of radial second Dirichlet eigenfunctions of fractional Schrödinger type operators of the form in the unit ball in with a nondecreasing radial potential . Specifically, we show that the eigenspace corresponding to the second radial eigenvalue is simple and spanned by an eigenfunction which changes sign precisely once in the radial variable and does not have zeroes anywhere else in . Moreover, by a new Hopf type lemma for supersolutions to a class of degenerate mixed boundary value problems, we show that has a nonvanishing fractional boundary derivative on . We apply this result to prove uniqueness and nondegeneracy of positive ground state solutions to the problem on , on . Here , and is strictly smaller than the critical Sobolev exponent.
Paper Structure (6 sections, 24 theorems, 149 equations)

This paper contains 6 sections, 24 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['main-theorem']}
  4. Nondegeneracy and uniqueness of ground state solutions
  5. Appendix I: A Hopf-type boundary point lemma
  6. Appendix II: Some topological lemmas on curve intersection

Key Result

Theorem 1.1

Suppose that, for some $q>\max(\frac{N}{2s},1)$ and $\beta >0$, Then $\sigma _2(V)$ is simple, and the associated eigenspace is spanned by an eigenfunction $w_2$ which changes sign exactly once in the radial variable. More precisely, there exists $r_0 \in (0,1)$ with the property that $w_2>0$ on $B_{r_0}$ and $w_2 < 0$ on $B \setminus \overline{B_{r_0}}$. Moreov where $\psi_{w_2}(1):= \liminf_{|x

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • ...and 38 more