Abstract:

In this paper, we are concerned with the multiplicity of solutions for the following fractional Laplacian problem $\begin{equation} \left\{\begin{array}{ll} { } (-{\Delta})^s{u}=\lambda|u|^{q-2}{u}+ \bigg(\int_\Omega\frac{|u(y)|^{2^\ast_{\mu, s}}}{|x-y|^\mu}{\rm d}y\bigg)|u|^{2^\ast_{\mu, s}-2}u, & x\in\Omega, \\ {u=0}, &x\in{{{\Bbb R}} ^N}\setminus\Omega \end{array}\right. \end{equation}$ where $\Omega\subset\mathbb{R} ^N$ is an open bounded set with continuous boundary, $N>2s$ with $s\in(0, 1)$, $\lambda$ is a real parameter, $\mu\in(0, N)$ and $q\in[2, 2^\ast_s)$, where $^\ast_{s}=\frac{2N}{N-2s}$, $^\ast_{\mu, s}=\frac{2N-\mu}{N-2s}$. Using Lusternik-Schnirelman theory, there exists $\bar{\lambda}>0$ such that for any $\lambda\in(0, \bar{\lambda})$, the problem has at least $cat_\Omega(\Omega)$ nontrivial solutions provided that $q=2$ and $N\geq4s$ or $q\in(2, 2^\ast_s)$ and $N>\frac{2s(q+2)}{q}$.

Key words: Choquard equation, Critical exponent, Lusternik-Schnirelman theory