Limiting Behavior of Constraint Minimizers for Inhomogeneous Fractional Schrödinger Equations
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917608721219584 |
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| author | Zhang, Hongfei Zhang, Shu |
| author_facet | Zhang, Hongfei Zhang, Shu |
| contents | This paper is devoted to the $L^2$-constraint variational problem \begin{equation*}
We study $L^2$-normalized solutions of the following inhomogeneous fractional Schrödinger equation \begin{equation*} (-Δ)^{s} u(x)+V(x)u(x)-a|x|^{-b}|u|^{2β^2}u(x)=μu(x)\ \ \mbox{in}\ \ \R^{N}. \end{equation*} Here $s\in(\frac{1}{2},1)$, $N>2s$, $a>0$, $0<b<\min\{\frac{N}{2},1\}$, $β=\sqrt{\frac{2s-b}{N}}$ and $V(x)\geq 0$ is an external potential. We get $L^2$-normalized solutions of the above equation by solving the associated constrained minimization problem. We prove that there exists a critical value $a^*>0$ such that minimizers exist for $0<a<a^*$, and minimizers do not exist for any $a>a^*$. In the case of $a=a^*$, one can obtain the classification results of the existence and non-existence for constraint minimizers, which are depended strongly on the value of $V(0)$. For $V(0)=0$, the limiting behavior of nonnegative minimizers is also analyzed when $a$ tend to $a^*$ from below. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_05834 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Limiting Behavior of Constraint Minimizers for Inhomogeneous Fractional Schrödinger Equations Zhang, Hongfei Zhang, Shu Analysis of PDEs This paper is devoted to the $L^2$-constraint variational problem \begin{equation*} We study $L^2$-normalized solutions of the following inhomogeneous fractional Schrödinger equation \begin{equation*} (-Δ)^{s} u(x)+V(x)u(x)-a|x|^{-b}|u|^{2β^2}u(x)=μu(x)\ \ \mbox{in}\ \ \R^{N}. \end{equation*} Here $s\in(\frac{1}{2},1)$, $N>2s$, $a>0$, $0<b<\min\{\frac{N}{2},1\}$, $β=\sqrt{\frac{2s-b}{N}}$ and $V(x)\geq 0$ is an external potential. We get $L^2$-normalized solutions of the above equation by solving the associated constrained minimization problem. We prove that there exists a critical value $a^*>0$ such that minimizers exist for $0<a<a^*$, and minimizers do not exist for any $a>a^*$. In the case of $a=a^*$, one can obtain the classification results of the existence and non-existence for constraint minimizers, which are depended strongly on the value of $V(0)$. For $V(0)=0$, the limiting behavior of nonnegative minimizers is also analyzed when $a$ tend to $a^*$ from below. |
| title | Limiting Behavior of Constraint Minimizers for Inhomogeneous Fractional Schrödinger Equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2302.05834 |