Three solutions for a fractional elliptic problem with asymmetric critical Choquard nonlinearity

Abstract

In this paper we study the existence and multiplicity of weak solutions for the following asymmetric nonlinear Choquard problem on fractional Laplacian: \begin{equation*} \begin{array}{rl} (-\Delta)^s u &= \displaystyle-\lambda|u|^{q-2}u + au + b\left( \int\limits_{\Omega} \frac{(u^{+}(y))^{2^{*}_{\mu ,s}}}{|x-y|^ \mu}\, dy\right) (u^{+})^{2^{*}_{\mu ,s}-2}u \quad\text{in} \; \Omega, u &= 0\quad \text{in} \; \mathbb{R}^{N}\backslash\Omega, \end{array} \end{equation*} where $\Omega$ is open bounded domain of $\mathbb{R}^{N}$ with $C^2$ boundary, $N > 2s$ and $s \in (0,1)$. Here $(-\Delta)^s$ is the fractional Laplace operator, $\lambda > 0$ is a real parameter, $q \in (1, 2)$, $a > 0$ and $b> 0$ are given constants, and $2^{*}_{\mu ,s} = \frac{2N-\mu}{N-2s}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and the notation $u^{+} = \max \{u, 0\}$. We prove that the above problem has at least three nontrivial solutions using the Mountain pass Lemma and Linking theorem.