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

Abstract

In this article, 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 [16]. 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. The method enables us to obtain new results on universal estimates of spatial singularities for elliptic equations and systems, under optimal growth assumptions that, unlike previous results, involve only the behavior of the nonlinearity at infinity. For semilinear systems of Lane-Emden type, these seem to be the first results on singularity estimates to cover the full subcritical range. In addition, we give an affirmative answer to the so-called Lane-Emden conjecture in three dimensions