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arXiv 2104.12997math.AP
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具有Sobolev临界指数和组合非线性项的薛定谔方程规范解的研究

Studies of normalized solutions to Schrödinger equations with Sobolev critical exponent and combined nonlinearities

Xinfu Li

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arXiv admin note: text overlap with arXiv:2104.09317
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摘要：

本文考虑具有组合非线性项\begin{equation*} \begin{cases} -Δu=λu+|u|^{2^*-2}u+μ|u|^{q-2}u,\ \ x\in\mathbb{R}^{N},\\ u\in H^1(\mathbb{R}^N),\ \int_{\mathbb{R}^N}|u|^2dx=a, \end{cases} \end{equation*}的Sobolev临界薛定谔方程，其中$N\geq 3$，$μ>0$，$λ\in \mathbb{R}$，$a>0$和$q\in (2,2^*)$ .本文证明了（1）当$γ_q:=\frac{N}{2}-\frac{N}{q}$和$K$为某个正常数时，$q\in (2,2+\frac{4}{N})$和$μa^{\frac{q(1-γ_q)}{2}}\leq (2K)^{\frac{qγ_q-2^*}{2^*-2}}$解的多重性和稳定性。该结果将Jeanjean等人在\cite{JEANJEAN-JENDREJ}和Jeanjean和Le \cite{Jeanjean-Le}中获得的关于$μa^{\frac{q(1-γ_q)}{2}}<(2K)^{\frac{qγ_q-2^*}{2^*-2}}$情况的结果扩展到$μa^{\frac{q(1-γ_q)}{2}}\leq (2K)^{\frac{qγ_q-2^*}{2^*-2}}$情况。(2)$q=2+\frac{4}{N}$和$μa^{\frac{q(1-γ_q)}{2}}\geq\bar{a}_N$的基态不存在，$\bar{a}_N$为某个正常数。我们对这个结果给出了一个不同于Wei和Wu \cite{Wei-Wu 2021}的新的证明.

英文摘要：

We consider the Sobolev critical Schrödinger equation with combined nonlinearities \begin{equation*} \begin{cases} -Δu=λu+|u|^{2^*-2}u+μ|u|^{q-2}u,\ \ x\in\mathbb{R}^{N},\\ u\in H^1(\mathbb{R}^N),\ \int_{\mathbb{R}^N}|u|^2dx=a, \end{cases} \end{equation*} where $N\geq 3$, $μ>0$, $λ\in \mathbb{R}$, $a>0$ and $q\in (2,2^*)$. We prove in this paper (1) Multiplicity and stability of solutions for $q\in (2,2+\frac{4}{N})$ and $μa^{\frac{q(1-γ_q)}{2}}\leq (2K)^{\frac{qγ_q-2^*}{2^*-2}}$ with $γ_q:=\frac{N}{2}-\frac{N}{q}$ and $K$ being some positive constant. This result extends the results obtained in Jeanjean et al. \cite{JEANJEAN-JENDREJ} and Jeanjean and Le \cite{Jeanjean-Le} for the case $μa^{\frac{q(1-γ_q)}{2}}<(2K)^{\frac{qγ_q-2^*}{2^*-2}}$ to the case $μa^{\frac{q(1-γ_q)}{2}}\leq (2K)^{\frac{qγ_q-2^*}{2^*-2}}$. (2) Nonexistence of ground states for $q=2+\frac{4}{N}$ and $μa^{\frac{q(1-γ_q)}{2}}\geq\bar{a}_N$ with $\bar{a}_N$ being some positive constant. We give a new proof to this result different with Wei and Wu \cite{Wei-Wu 2021}.

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。

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