This paper focuses on static solutions for the following Choquard equation with zero mass and Coulomb potential \(\begin{aligned} -\Delta u+\left( \frac{1}{4\pi |x|}*u^2\right) u=\mu |u|^{p-2}u+(I_{\alpha }*|u|^{\alpha +3})|u|^{\alpha +1}u, \ \ \ \ x\in {\mathbb {R}}^{3}, \end{aligned}\) where \(\mu >0\) , \(\frac{18}{7}<p\le 6\) , \(\alpha \in (0, 3)\) , \(\alpha +3\) is the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality, \(I_{\alpha }{:}\,\mathbb {R}^3\rightarrow \mathbb {R}\) is the Riesz potential, and \(\frac{1}{4\pi |x|}\) is the Coulomb potential. By carefully analyzing the intricate interplay between the power and Coulomb terms, we establish three types of variational geometries of the problem and prove the following existence results based on the behavior of p: (1) the existence of two solutions, one being a local minimizer and the other of mountain-pass type, for an explicit range \(0<\mu <\mathrm {Const.}\) when \(\frac{18}{7}<p<3\) ;
(2) the existence of a positive solution if \(\mu \) takes some particular value when \(p=3\) ;
(3) the existence of a ground state solution for all \(\mu >0\) when \(4<p<6\) , and for two explicit ranges \(\mu >\mathrm {Const.}\) when \(3<p<4\) and \(p=4\) .
Furthermore, we obtain a non-existence result for the case \(p=6\) . Particularly, we identify different compactness thresholds for above three cases, and introduce three types of test functions to control the corresponding minimax levels to be less than prescribed thresholds, thereby overcoming the loss of compactness arising from the nonlocal critical term. The derivation of these strict inequalities is a novel contribution and constitutes one of the noteworthy highlights of this work, which is available and new for the limiting Sobolev critical problem as \(\alpha \rightarrow 0\) . We believe that the underlying ideas have potential for future development and can be applied to a broader range of variational problems with critical growth.
