Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part I: Elliptic equations and systems

In this paper, we study some new connections between Liouville-type theorems and local properties of nonnegative solutions to superlinear elliptic problems. Namely, we develop a general method for derivation of universal, pointwise a priori estimates of local solutions from Liouville-type theorems, which provides a simpler and unified treatment for such questions. The method is based on rescaling arguments combined with a key “doubling” property, and it is different from the classical rescaling method of Gidas and Spruck. As an important heuristic consequence of our approach, it turns out that universal boundedness theorems for local solutions and Liouville-type theorems are essentially equivalent. ∗Supported in part by NSF Grant DMS-0400702 †Supported in part by VEGA Grant 1/3021/06