Existence and uniqueness of solutions for Choquard equation involving Hardy–Littlewood–Sobolev critical exponent

Abstract

In this paper, we first prove that each positive solution of

$$\begin{aligned} -\Delta u=\big (I_{\alpha }*|u|^{2_{\alpha }^{*}}\big )|u|^{2_{\alpha }^*-2}u, \quad u\in \mathcal {D}^{1,2}({\mathbb {R}}^{N}) \end{aligned}$$

is radially symmetric, monotone decreasing about some point and has the form

$$\begin{aligned} c_\alpha \left( \frac{t}{t^2+|x-x_0|^2}\right) ^{\frac{N-2}{2}}, \end{aligned}$$

where $0<\alpha <N$ if $N=3$ or 4, and $N-4\le \alpha <N$ if $N\ge 5$, ${2_{\alpha }^{*}}:=\frac{N+\alpha }{N-2}$ is the upper Hardy–Littlewood–Sobolev critical exponent, $t>0$ is a constant and $c_\alpha >0$ depends only on $\alpha$ and N. Based on this uniqueness result, we then study the following nonlinear Choquard equation

$$\begin{aligned} -\Delta u+V(x)u=\left( I_{\alpha }*|u|^{2_{\alpha }^{*}}\right) |u|^{2_{\alpha }^*-2}u, \quad u\in \mathcal {D}^{1,2}({\mathbb {R}}^{N}). \end{aligned}$$

By using Lions’ Concentration-Compactness Principle, we obtain a global compactness result, i.e. we give a complete description for the Palais–Smale sequences of the corresponding energy functional. Adopting this description, we are succeed in proving the existence of at least one positive solution if $\Vert V(x)\Vert _{L^\frac{N}{2}}$ is suitable small. This result generalizes the result for semilinear Schrödinger equation by Benci and Cerami (J Funct Anal 88:90–117, 1990) to Choquard equation.