Multiple solutions for nonhomogeneous Choquard equation involving Hardy–Littlewood–Sobolev critical exponent

Abstract

We consider the following critical nonhomogeneous Choquard equation

$$\begin{aligned} -\Delta u =\left( \mathop {\int }\limits _{\Omega }\frac{|u(y)|^{2_{\mu }^{*}}}{|x-y|^{\mu }}\hbox {d}y\right) |u|^{2_{\mu }^{*}-2}u+\lambda u+f(x) \quad \hbox {in}\quad \Omega , \end{aligned}$$

where $\Omega$ is a smooth bounded domain of $\mathbb {R}^N$, 0 in interior of $\Omega$, $\lambda \in \mathbb {R}$, $N\ge 7$, $0<\mu <N$, $2_{\mu }^{*}=(2N-\mu )/(N-2)$ is the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality, and f(x) is a given function. Using variational methods, we obtain the existence of multiple solutions for the above problem when $0<\lambda <\lambda _{1}$, where $\lambda _{1}$ is the first eigenvalue of $-\Delta$ in $H_{0}^{1}(\Omega )$.