分数阶Choquard方程基态解的存在性与集中性
Existence and concentration of ground states to fractional Choquard equations
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中文总结 AI 辅助
本文研究一类分数阶Choquard方程，在势函数满足适当假设时，得到其基态解的存在性、集中性及极限行为，为分数阶Choquard方程的基态解研究提供了理论结果。
中文摘要 AI 辅助
本文研究非线性分数阶Choquard方程：\begin{equation}\label{eq:0.1a} (-\Delta)^su+Vu=(|x|^{-\gamma}*|u|^2)u \quad {\rm in} \quad \mathbb{R}^N, \end{equation}其中$0<\gamma<4$，$0<s<1$，$N\geq 4$，$V\in C^1(\mathbb{R}^N)$为正势函数。令$s_0=\frac{\gamma}{4}$，在$V$满足适当假设下，证明当$s\in(s_0,1)$时方程存在非负基态解，而当$0<s\leq s_0$时不存在基态解；还证明任一基态解$u_s$在$s\downarrow s_0$时爆破并集中于$V$的极小值点，且沿子列当$s\uparrow 1$时$u_s$收敛到经典Choquard方程的基态解。
英文摘要
In this paper, we study the nonlinear fractional Choquard equation \begin{equation}\label{eq:0.1a} (-Δ)^su+Vu=(|x|^{-γ}*|u|^2)u \quad {\rm in} \quad \mathbb{R}^N, \end{equation} where $0<γ<4$, $0<s<1$, $N\geq 4$ and $V\in C^1(\mathbb{R}^N)$ is a positive potential. Set $s_0=\frac γ{4}$. Under suitable assumptions on $V$, we prove that the equation admits a nonnegative ground state solution for $s\in(s_0,1)$, whereas no ground state solution exists for $0<s\le s_0$. Furthermore, we show that any ground state solution $u_s$ blows up and concentrates at a minimum point of $V$ as $s\downarrow s_0$. Finally, up to a subsequence, the ground state solution $u_s$ converges to a ground state solution of the classical Choquard equation as $s\uparrow 1$.
