Method of scaling spheres: Liouville theorems in inner or outer (unbounded or bounded) generalized radially convex domains, blowing-up analysis on domains with not necessarily $C^1$ boundary and other applications

Abstract

In this paper, we aim to introduce the method of scaling spheres (MSS) as a unified approach to Liouville theorems on general domains in $\mathbb R^n$, and apply it to establish Liouville theorems on arbitrary unbounded or bounded MSS applicable domains for general ($\leq n$-th order) PDEs and integral equations without translation invariance or with singularities. The set of MSS applicable domains includes any unbounded or bounded generalized radially convex domains and any complementary sets of their closures, which is invariant under Kelvin transforms and is the maximal collection of simply connected domains such that the MSS works. For instance, $ \mathbb R^n$, $\mathbb R^n_+$, balls, cone-like domains, convex domains, star-shaped domains and all the complements of their closures are MSS applicable domains. MSS applicable domains is to the MSS what convex domains is to the method of moving planes. As applications, we derive a priori estimates and existence of solutions from the boundary H\"{o}lder estimates for Dirichlet or Navier problems of Lane-Emden equations by applying the blowing-up argument on domains with blowing-up cone boundary (BCB domains for short). After the blowing-up procedure, the BCB domains allow the limiting shape of the domain to be a cone (half space is a cone). While the classical blowing-up techniques in previous papers work on $C^1$-smooth domains, we are able to apply blowing-up analysis on more general BCB domains on which the boundary H\"{o}lder estimates hold (can be guaranteed by uniform exterior cone property etc). Since there are no smoothness conditions on the boundary of MSS applicable domain and BCB domain, our results clearly reveal that the existence and nonexistence of solutions mainly rely on topology (not smoothness) of the domain.