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.