Nonlinear Liouville-type theorems and singularity estimates in semilinear and quasilinear elliptic problems

Philippe Souplet
Université Paris-Nord, France

Abstract: We study some new connections between elliptic Liouville-type theorems and local properties of nonnegative classical solutions to superlinear elliptic problems. Namely, we develop a general method for derivation of universal, pointwise a priori estimates from Liouville-type theorems. Our method, which is based on rescaling arguments combined with a key .doubling. property. It is different from the classical rescaling method of Gidas and Spruck.

New results on universal estimates of singularities for local solutions of elliptic equations and systems will be presented. As another consequence, we give an affirmative answer to the so-called Lane-Emden conjecture in three dimensions. As a heuristic consequence of our approach, it turns out that universal boundedness theorems for local solutions and Liouville-type theorems are essentially equivalent. This approach also has interesting consequences for parabolic problems, which I will briefly mention.

Joint work with Peter Polacik and Pavol Quittner.