Concentration phenomenon of solutions for fractional Choquard equations with upper critical growth

Abstract

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

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