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

Abstract

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 H 1 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 p -Laplace equation. We use ideas from the potential theories for the p -Laplace equations and Hessian equations.