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