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非自治Choquard方程的归一化基态解

Normalized ground states solutions for nonautonomous Choquard equations

Huxiao Luo, Lushun Wang

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摘要：

本文研究了非自治Choquard方程$$-Δu-λu=\left(\frac{1}{|x|^μ}\ast A|u|^{p}\right)A|u|^{p-2}u,\quad \int_{\mathbb{R}^{N}}|u|^{2}dx=c,\quad u\in H^1(\mathbb{R}^N,\mathbb{R}),$$的归一化基态解，其中$c>0$，$0<μ<N$，$λ\in\mathbb{R}$，$A\in C^1(\mathbb{R}^N,\mathbb{R})$ .对于$p\in(2_{*,μ}, \bar{p})$，我们证明了Choquard方程具有基态正规化解，并且基态集合是轨道稳定的.对于$p\in (\bar{p},2^*_μ)$，我们找到了一个规范化的解决方案，这是不是一个全局极小。$2^*_μ$和$2_{*,μ}$分别是Hardy-Littlewood-Sobolev不等式的上临界指数和下临界指数。$\bar{p}$是$L^2-$临界指数。所得结果推广和改进了相关结果.

英文摘要：

In this paper, we study normalized ground state solutions for the following nonautonomous Choquard equation: $$-Δu-λu=\left(\frac{1}{|x|^μ}\ast A|u|^{p}\right)A|u|^{p-2}u,\quad \int_{\mathbb{R}^{N}}|u|^{2}dx=c,\quad u\in H^1(\mathbb{R}^N,\mathbb{R}),$$ where $c>0$, $0<μ<N$, $λ\in\mathbb{R}$, $A\in C^1(\mathbb{R}^N,\mathbb{R})$. For $p\in(2_{*,μ}, \bar{p})$, we prove that the Choquard equation possesses ground state normalized solutions, and the set of ground states is orbitally stable. For $p\in (\bar{p},2^*_μ)$, we find a normalized solution, which is not a global minimizer. $2^*_μ$ and $2_{*,μ}$ are the upper and lower critical exponents due to the Hardy-Littlewood-Sobolev inequality, respectively. $\bar{p}$ is $L^2-$critical exponent. Our results generalize and extend some related results.

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