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arXiv 2602.16263math.AP
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非线性Schrödinger方程在有界域中涉及混合非线性的解的存在性和不存在性

Existence and nonexistence of normalized solutions for nonlinear Schrödinger equation involving combined nonlinearities in bounded domain

Shanxi Normal University（山西师范大学）；Zhejiang Normal University（浙江师范大学）

Zhen-Feng Jin, Weimin Zhang

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AI总结：

本文研究了非线性Schrödinger方程在有界域中混合非线性的解的存在性和不存在性，通过能量极小值、单调性技巧、移动平面法等方法，探讨了不同条件下解的性质。

摘要：

本文考虑以下方程的解的存在性、多重性和不存在性：\begin{equation*} \begin{cases} \begin{aligned} &-Δu+ωu=μu^{p-1}+u^{q-1},~ u>0 \quad &&\text { in } Ω, \\ &u=0 &&\text { on } \partialΩ, \\ \end{aligned} \end{cases} \end{equation*}其中给定$ L^2 $-范数$\|u\|_2^2=ρ$，其中$ N\ge 1 $，$ ρ>0 $，$ μ\in \mathbb{R} $，$ 1<p\le q $，且$ Ω\subset\mathbb{R}^N $是有界光滑域。参数$ ω\in\mathbb{R} $作为拉格朗日乘数出现。首先，当$ 2<p\le q\le \frac{2N}{(N-2)^+} $且$ ρ $很小时，我们建立了能量的局部极小值解的存在性。进一步，当$ μ\ge 0 $且$ Ω $是星形域时，利用单调性技巧和Pohozaev恒等式，我们证明存在一个山 pass 型的第二个解。其次，当$ μ\ge 0 $，$ N\ge 3 $，$ 1<p\le 2 $，$ q\ge \max\left\{\frac{2N}{N-2}, 3\right\} $且$ Ω $是凸域时，利用移动平面法，我们证明对于大的$ ρ $，规范化解不存在。最后，当$ μ=0 $，$ N\ge 3 $，$ q=\frac{2N}{N-2} $且$ Ω $是球体时，通过连续性论证，我们给出了Brézis-Nirenberg问题规范化解的二元性结果。

英文摘要：

In this paper, we consider the existence, multiplicity and nonexistence of solutions for the following equation \begin{equation*} \begin{cases} \begin{aligned} &-Δu+ωu=μu^{p-1}+u^{q-1},~ u>0 \quad &&\text { in } Ω, \\ &u=0 &&\text { on } \partialΩ, \\ \end{aligned} \end{cases} \end{equation*} with prescribed $L^2$-norm $\|u\|_2^2=ρ$, where $N\ge 1$, $ρ>0$, $μ\in \mathbb{R}$, $1<p\le q$, and $Ω\subset\mathbb{R}^N$ is a bounded smooth domain. The parameter $ω\in\mathbb{R}$ arises as a Lagrange multiplier. Firstly, when $2<p\le q\le \frac{2N}{(N-2)^+}$ and $ρ$ is small, we establish the existence of a local minimizer of energy. Furthermore, when $μ\ge 0$ and $Ω$ is a star-shaped domain, using the monotonicity trick and the Pohozaev identity, we show that there exists a second solution which is of mountain pass type. Secondly, when $μ\ge 0$, $N\ge 3$, $1<p\le 2$, $q\ge \max\left\{\frac{2N}{N-2}, 3\right\}$ and $Ω$ is a convex domain, using the moving-plane method, we prove the nonexistence of normalized solutions for large $ρ$. Finally, when $μ=0$, $N\ge 3$, $q=\frac{2N}{N-2}$ and $Ω$ is a ball, we give a dichotomy result of normalized solutions for the Brézis-Nirenberg problem by continuation arguments.

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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