Concentration phenomenon of solutions for fractional Choquard equations with upper critical growth,Fractional Calculus and Applied Analysis

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Concentration phenomenon of solutions for fractional Choquard equations with upper critical growth
Fractional Calculus and Applied Analysis ( IF 3 ) Pub Date : 2022-06-02 , DOI: 10.1007/s13540-022-00052-0
Quanqing Li , Meiqi Liu , Houwang Li

In this article, we focus on the following fractional Choquard equation involving upper critical exponent

$$\begin{aligned} \varepsilon ^{2s}(-\varDelta )^su+V(x)u=P(x)f(u)+\varepsilon ^{\mu -N}Q(x)[|x|^{-\mu }*|u|^{2_{\mu ,s}^*}]|u|^{2_{\mu ,s}^*-2}u, \ x \in {\mathbb {R}}^N, \end{aligned}$$

where $\varepsilon >0$, $0<s<1$, $(-\varDelta )^s$ denotes the fractional Laplacian of order s, $N>2s$, $0<\mu <N$ and $2_{\mu ,s}^*=\frac{2N-\mu }{N-2s}$. Under suitable assumptions on the potentials V(x), P(x) and Q(x), we obtain the existence and concentration of positive solutions and prove that the semiclassical solutions $w_\varepsilon $ with maximum points $x_\varepsilon $ concentrating at a special set ${\mathcal {S}}_p$ characterized by V(x), P(x) and Q(x). Furthermore, for any sequence $x_\varepsilon \rightarrow x_0 \in {\mathcal {S}}_p$, $v_\varepsilon (x):=w_\varepsilon (\varepsilon x+x_\varepsilon )$ converges in $H^s({\mathbb {R}}^N)$ to a ground state solution v of

$$\begin{aligned} (-\varDelta )^sv+V(x_0)v=P(x_0)f(v)+Q(x_0)[|x|^{-\mu }*|v|^{2_{\mu ,s}^*}]|v|^{2_{\mu ,s}^*-2}v, \ x \in {\mathbb {R}}^N. \end{aligned}$$

中文翻译：

具有上临界增长的分数阶Choquard方程解的集中现象

在本文中，我们关注以下涉及上临界指数的分数 Choquard 方程

$$\begin{对齐} \varepsilon ^{2s}(-\varDelta )^su+V(x)u=P(x)f(u)+\varepsilon ^{\mu -N}Q(x)[ |x|^{-\mu }*|u|^{2_{\mu ,s}^*}]|u|^{2_{\mu ,s}^*-2}u, \ x \in { \mathbb {R}}^N, \end{对齐}$$

其中$\varepsilon >0$ , $0<s<1$ , $(-\varDelta )^s$表示s阶的小数拉普拉斯算子, $N>2s$ , $0 <\mu <N$和$2_{\mu ,s}^*=\frac{2N-\mu }{N-2s}$。在对势V ( x )、P ( x ) 和Q ( x ) 的适当假设下，我们得到了正解的存在性和集中性，并证明了具有最大点$x_ \ varepsilon $专注于一个特殊的集合${\mathcal {S}}_p$以V ( x )、P ( x ) 和Q ( x ) 为特征。此外，对于任何序列$x_\varepsilon \rightarrow x_0 \in {\mathcal {S}}_p$，$v_\varepsilon (x):=w_\varepsilon (\varepsilon x+x_\varepsilon )$收敛于$H^s({\mathbb {R}}^N)$到基态解v