Asymptotic behavior of ground states for a fractional Choquard equation with critical growth

Abstract

In this paper, we are concerned with the following fractional Choquard equation with critical growth: $$(-\Delta)^s u+\lambda V(x)u=(|x|^{-\mu} \ast F(u))f(u)+|u|^{2^*_s-2}u ~\hbox{in}~\mathbb{R}^N,$$ where $s\in (0,1)$, $N>2s$, $\mu\in (0,N)$, $2^*_s=\frac{2N}{N-2s}$ is the fractional critical exponent, $V$ is a steep well potential, $F(t)=\int_0^tf(s)ds$. Under some assumptions on $f$, the existence and asymptotic behavior of the positive ground states are established. In particular, if $f(u)=|u|^{p-2}u$, we obtain the range of $p$ when the equation has the positive ground states for three cases $2s4s$.

Keywords

• fractional choquard equation
• critical growth
• ground states
• asymptotic behavior