Uniqueness and Nondegeneracy of ground states of $ -Δu + (-Δ)^s u+u = u^{p+1} \quad \hbox{in $\mathbb{R}^n$}$ when $s$ is close to $0$ and $1$
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866929603461775360 |
|---|---|
| author | Su, Xifeng Zhang, Chengxiang Zhang, Jiwen |
| author_facet | Su, Xifeng Zhang, Chengxiang Zhang, Jiwen |
| contents | We are concerned with the mixed local/nonlocal Schrödinger equation
\begin{equation}
- Δu + (-Δ)^s u+u = u^{p+1} \quad \hbox{in $\mathbb{R}^n$,}
\end{equation}
for arbitrary space dimension $n\geqslant1$, $s\in(0,1)$, and $p\in(0,2^*-2)$ with $2^*$ the critical Sobolev exponent.
We provide the existence and several fundamental properties of nonnegative solutions for the above equation. And then, we prove that, if $s$ is close to $0$ and $1$, respectively, such equation then possesses a unique (up to translations) ground state, which is nondegenerate. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_19616 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniqueness and Nondegeneracy of ground states of $ -Δu + (-Δ)^s u+u = u^{p+1} \quad \hbox{in $\mathbb{R}^n$}$ when $s$ is close to $0$ and $1$ Su, Xifeng Zhang, Chengxiang Zhang, Jiwen Analysis of PDEs 35A02, 35B65, 35J10, 35R11 We are concerned with the mixed local/nonlocal Schrödinger equation \begin{equation} - Δu + (-Δ)^s u+u = u^{p+1} \quad \hbox{in $\mathbb{R}^n$,} \end{equation} for arbitrary space dimension $n\geqslant1$, $s\in(0,1)$, and $p\in(0,2^*-2)$ with $2^*$ the critical Sobolev exponent. We provide the existence and several fundamental properties of nonnegative solutions for the above equation. And then, we prove that, if $s$ is close to $0$ and $1$, respectively, such equation then possesses a unique (up to translations) ground state, which is nondegenerate. |
| title | Uniqueness and Nondegeneracy of ground states of $ -Δu + (-Δ)^s u+u = u^{p+1} \quad \hbox{in $\mathbb{R}^n$}$ when $s$ is close to $0$ and $1$ |
| topic | Analysis of PDEs 35A02, 35B65, 35J10, 35R11 |
| url | https://arxiv.org/abs/2410.19616 |