Regularity, symmetry, and uniqueness of some integral type quasilinear equations

We study integral equations corresponding to some quasilinear equations with nonlinearities of Lane–Emden and Hartree type. Regularity, symmetry, and uniqueness of these equations are considered. We obtain the uniqueness of the ground state of H1H1 critical Hartree equation and extend the moving plane method of integral equation in [W. Chen, C. Li, B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math. LIX (2006) 0330–0343; W. Chen, C. Li, B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations 30 (1–3) (2005) 59–65] to some integral equations corresponding to the pp-Laplace equation. We use ideas from the potential theories for the pp-Laplace equations and Hessian equations.