Existence of solutions for quasi-linear Hartree equations with critical growth

In this paper, we study the existence of nontrivial solutions for a class of quasilinear critical elliptic equation: [Formula: see text] where [Formula: see text] denotes the [Formula: see text]-Laplace operator, [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] is of critical growth due to the Hardy–Littlewood–Sobolev inequality. First, for [Formula: see text] is a bounded domain and [Formula: see text], by using the decay estimate of the extremal functions of the quasilinear critical Hartree equation, we are able to obtain the existence of nontrivial solutions for the quasilinear Brézis–Nirenberg-type problem. Second, for the problem set in [Formula: see text], we obtain the existence of nontrivial solutions for the quasilinear critical Hartree equation under the effect of potential function by establishing a nonlocal version of the second concentration-compactness principle for [Formula: see text]-Laplacian.