Ground state solution for Schrödinger-Choquard equation: doubly critical case

In this paper, we investigate the following Schrödinger-Choquard equation: $$ -\Delta u+u = (I_{\alpha }*|u|^{2_{\alpha }^{\sharp }})|u|^{2_{\alpha }^{ \sharp }-2}u + |u|^{q-2}u + |u|^{r-2}u, \quad x\in \mathbb{R}^{N}, $$ where $N\geqslant 1$, $\alpha \in (0,N)$, $q\in (2,2+\frac{4}{N})$ and $r\in [2+\frac{4}{N},2^{*}]$. Here, $I_{\alpha }$ denotes the Riesz potential, $2_{\alpha }^{\sharp }=\frac{N+\alpha }{N}$ represents the lower critical exponent in the context of the Hardy-Littlewood-Sobolev inequality, and 2∗ is the Sobolev critical exponent. By means of the Lions-type theorem, Lieb’s translation lemma, and the Nehari manifold, we establish the existence of non-negative ground-state solutions to the aforementioned equation for $N\geqslant 5$ and $N=1,2,3,4$, respectively. Our results extend those in the relevant literature.