关键词: 分数阶Choquard方程, 临界增长, 超临界增长, 截断技巧

Abstract:

We consider the following fractional Choquard equation with critical or supercritical growth $(-\triangle)^s u+u=f(u)+\lambda(|x|^{-\mu} \ast|u|^q\big)|u|^{q-2}u, x\in \Omega, $ where $s \in (0, 1)$, $\mu\in (0, N)$, $N>2s$, $q\geq 2_{\mu, s}^\ast$, $f$ is a continuous function. It is well-known that $2_{\mu, s}^\ast=\frac{2N-\mu}{N-2s}$ and $2_{\mu, s}=\frac{2N-\mu}{N}$ are critical exponents for the above equation in the sense of Hardy-Littlewood-Sobolev inequality. Many existence results have been established for $q \in[2_{\mu, s}, 2_{\mu, s}^\ast]$ in recent years. Here we are interested in critical or supercritical case for the above equation. Under some assumptions of $f$, the existence and multiplicity of solutions for the above equation can be obtained by applying some analytical techniques.

Key words: Fractional Choquard equation, Critical, Supercritical growth, Truncation technique