Qualitative properties of positive solutions of a mixed order nonlinear Schrödinger equation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912120396840960 |
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| author | Dipierro, Serena Su, Xifeng Valdinoci, Enrico Zhang, Jiwen |
| author_facet | Dipierro, Serena Su, Xifeng Valdinoci, Enrico Zhang, Jiwen |
| contents | In this paper, we deal with the following mixed local/nonlocal Schrödinger equation
\begin{equation*}
\left\{
\begin{array}{ll}
- Δu + (-Δ)^s u+u = u^p \quad \hbox{in $\mathbb{R}^n$,}
u>0 \quad \hbox{in $\mathbb{R}^n$,}
\lim\limits_{|x|\to+\infty}u(x)=0,
\end{array}
\right.
\end{equation*} where $n\geqslant2$, $s\in (0,1)$ and $p\in\left(1,\frac{n+2}{n-2}\right)$.
The existence of positive solutions for the above problem is proved, relying on some new regularity results. In addition, we study the power-type decay and the radial symmetry properties of such solutions.
The methods make use also of some basic properties of the heat kernel and the Bessel kernel associated with the operator $- Δ+ (-Δ)^s$: in this context, we provide self-contained proofs of these results based on Fourier analysis techniques. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09941 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Qualitative properties of positive solutions of a mixed order nonlinear Schrödinger equation Dipierro, Serena Su, Xifeng Valdinoci, Enrico Zhang, Jiwen Analysis of PDEs 35A08, 35B06, 35B09, 35B40, 35J10 In this paper, we deal with the following mixed local/nonlocal Schrödinger equation \begin{equation*} \left\{ \begin{array}{ll} - Δu + (-Δ)^s u+u = u^p \quad \hbox{in $\mathbb{R}^n$,} u>0 \quad \hbox{in $\mathbb{R}^n$,} \lim\limits_{|x|\to+\infty}u(x)=0, \end{array} \right. \end{equation*} where $n\geqslant2$, $s\in (0,1)$ and $p\in\left(1,\frac{n+2}{n-2}\right)$. The existence of positive solutions for the above problem is proved, relying on some new regularity results. In addition, we study the power-type decay and the radial symmetry properties of such solutions. The methods make use also of some basic properties of the heat kernel and the Bessel kernel associated with the operator $- Δ+ (-Δ)^s$: in this context, we provide self-contained proofs of these results based on Fourier analysis techniques. |
| title | Qualitative properties of positive solutions of a mixed order nonlinear Schrödinger equation |
| topic | Analysis of PDEs 35A08, 35B06, 35B09, 35B40, 35J10 |
| url | https://arxiv.org/abs/2411.09941 |