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In this paper, we aim to introduce the method of scaling spheres (MSS) as aunified approach to Liouville theorems on general domains in $\mathbb R^n$, andapply it to establish Liouville theorems on arbitrary unbounded or bounded MSSapplicable domains for general ($\leq n$-th order) PDEs and integral equationswithout translation invariance or with singularities. The set of MSS applicabledomains includes any unbounded or bounded generalized radially convex domainsand any complementary sets of their closures, which is invariant under Kelvintransforms and is the maximal collection of simply connected domains such thatthe MSS works. For instance, $ \mathbb R^n$, $\mathbb R^n_+$, balls, cone-likedomains, convex domains, star-shaped domains and all the complements of theirclosures are MSS applicable domains. MSS applicable domains is to the MSS whatconvex domains is to the method of moving planes. As applications, we derive apriori estimates and existence of solutions from the boundary H\"{o}lderestimates for Dirichlet or Navier problems of Lane-Emden equations by applyingthe blowing-up argument on domains with blowing-up cone boundary (BCB domainsfor short). After the blowing-up procedure, the BCB domains allow the limitingshape of the domain to be a cone (half space is a cone). While the classicalblowing-up techniques in previous papers work on $C^1$-smooth domains, we areable to apply blowing-up analysis on more general BCB domains on which theboundary H\"{o}lder estimates hold (can be guaranteed by uniform exterior coneproperty etc). Since there are no smoothness conditions on the boundary of MSSapplicable domain and BCB domain, our results clearly reveal that the existenceand nonexistence of solutions mainly rely on topology (not smoothness) of thedomain.

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