Existence of Ground State Solutions for the Kirchhoff–Choquard Equation With Critical Exponent and External Coulomb Potential

Abstract

We study the critical Kirchhoff–Choquard type equation $$-(a+b{\int }_{{\mathbb{R}}^3}|\mathrm{\nabla }u{|}^2\mathrm{d}x)\mathrm{△}u+(\beta -\frac{\mu }{|x|})u=(I_{\alpha }*|u{|}^q)|u{|}^{q-2}u,u\in H^1({\mathbb{R}}^3)$$ where $\alpha \in (0,3)$ , $b\ge 0$ , $a$ , $\beta$ and $\mu$ are positive constants, $I_{\alpha }$ denotes the Riesz potential and $q=\frac{3+\alpha }{3}$ is the lower critical Hardy–Littlewood–Sobolev exponent. By means of the compactness-concentration principle and the squeezing energy inequality, we obtain the existence of ground state solutions of Nehari–Pohoz̆aev type for the above problem for sufficiently small $\mu >0$ .