Liouville type theorems, a priori estimates and existence of solutions for critical and super-critical order Hardy–Hénon type equations in Rⁿ

In this paper, we first consider the critical order Hardy–Hénon type equations and inequalities (-Δ)n2u(x)≥up(x)|x|a,x∈Rn,where n≥ 4 is even, - ∞< a< n, and 1 < p< + ∞. We prove Liouville theorems (Theorems 1.1 and 1.3), that is, the unique nonnegative solution is u≡ 0. Then as an immediate application, by applying method of moving planes in a local way, blowing-up techniques and the Leray–Schauder fixed point theorem, we derive a priori estimates and hence existence of positive solutions to critical order Lane–Emden equations in bounded domains (Theorems 1.4 and 1.6). Extensions to super-critical order Hardy–Hénon type equations and inequalities will also be included (Theorems 1.8 and 1.11). In critical and super-critical order Hardy–Hénon type inequalities, there are no growth conditions on the nonlinearity f(x, u) w.r.t. u and hence the nonlinear term can grow exponentially (or even faster) on u (Theorems 1.3, 1.8 and Remark 1.10).