Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.02120 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915568159817728 |
|---|---|
| author | Fall, Mouhamed Moustapha Weth, Tobias |
| author_facet | Fall, Mouhamed Moustapha Weth, Tobias |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02120 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Second radial eigenfunctions to a fractional Dirichlet problem and uniqueness for a semilinear equation Fall, Mouhamed Moustapha Weth, Tobias Analysis of PDEs 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. |
| title | Second radial eigenfunctions to a fractional Dirichlet problem and uniqueness for a semilinear equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.02120 |