Ground state solutions of Pohoz̆aev type for the Choquard equation with external Coulomb potential and critical exponent

Abstract

This paper is concerned with the following Choquard equation $\begin{array}{c}-△u+(\omega -\frac{\beta }{\left|x\right|})u=(I_{\alpha }\ast {\left|u\right|}^{\frac{N+\alpha }{N-2}}){\left|u\right|}^{\frac{N+\alpha }{N-2}-2}u,x\in {\mathbb{R}}^N,\hfill \end{array}$ where $N\ge 3$, $\alpha \in (0,N)$, $I_{\alpha }$ is the Riesz potential, $\omega$ and $\beta$ are positive real numbers. For $\beta =0$, it is known that the above equation has no nontrivial solution. For $\beta >0$, $-\beta ∕\left|x\right|$ is called the external Coulomb potential. If $0<\beta <c(\alpha ,\omega ,N)$, we obtain a ground state solution of Pohoz̆aev type for the above equation, where $c(\alpha ,\omega ,N)$ is a constant that can be expressed explicitly via $\alpha$, $\omega$ and $N$.