arXiv 2312.159172023-12-26math.AP

含势函数与Sobolev临界指数的非线性薛定谔方程的规范化解

Normalized solutions for nonlinear Schrödinger equation involving potential and Sobolev critical exponent

Zhen-Feng Jin, Weimin Zhang

AI总结：

本文研究含势函数与Sobolev临界指数的非线性薛定谔方程在给定L²范数下正解的存在性，通过两种Pohozaev恒等式获得解的存在性，推广了Wei和Wu的结果至势函数情形。

摘要：

本文考虑如下含势函数与Sobolev临界指数的非线性薛定谔方程具有给定$L^2$-范数的正解的存在性：\n\begin{equation*}\n\begin{cases}\n-\Delta u+V(x)u=\lambda u+\mu|u|^{p-2}u+|u|^{\frac{4}{N-2}}u \\;\\; \text { in } \mathbb{R}^N, \n\\|u\\|_2=a>0,\n\end{cases}\n\end{equation*}\n其中$N\ge 3$，$\mu>0$，$p\in [2+\frac{4}{N}, \frac{2N}{N-2})$且$V\in C^1(\mathbb{R}^N)$。在关于$V$的不同假设下，我们推导出两个不同的Pohozaev恒等式。基于这两种情形，我们分别得到了正解的存在性。据我们所知，我们未发现任何关于具有Sobolev临界增长和势函数$V \not\equiv 0$的规范化解的研究。我们的结果将Wei和Wu [J. Funct. Anal. 283(2022)]的一些结果推广到了含势函数的情形。

英文摘要：

In this paper, we consider the existence of positive solutions with prescribed $L^2$-norm for the following nonlinear Schrödinger equation involving potential and Sobolev critical exponent \begin{equation*} \begin{cases} -Δu+V(x)u=λu+μ|u|^{p-2}u+|u|^{\frac{4}{N-2}}u \;\;\text { in } \mathbb{R}^N, \\ \|u\|_2=a>0,\\ \end{cases} \end{equation*} where $N\ge 3$, $μ>0$, $p\in [2+\frac{4}{N}, \frac{2N}{N-2})$ and $V\in C^1(\mathbb{R}^N)$. Under different assumptions on $V$, we derive two different Pohozaev identities. Based on these two cases, we respectively obtain the existence of positive solution. As far as we are aware, we did not find any works on normalized solutions with Sobolev critical growth and potential $V \not\equiv 0$. Our results extend some results of Wei and Wu [J. Funct. Anal. 283(2022)] to the potential case.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

AI总结：

摘要：

英文摘要：

相关论文