Open AccessUniformly elliptic Liouville type equations: concentration compactness and a priori estimates
Author(s) -
Daniele Bartolucci,
Luigi Orsina
Publication year - 2005
Publication title -
communications on pure and applied analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.077
H-Index - 42
eISSN - 1553-5258
pISSN - 1534-0392
DOI - 10.3934/cpaa.2005.4.499
Subject(s) - mathematics , compact space , bounded function , type (biology) , a priori and a posteriori , mathematical analysis , dirichlet problem , uniform boundedness , dirichlet distribution , pure mathematics , boundary value problem , ecology , philosophy , epistemology , biology
We analyze the singular behavior of the Green's function for uniformly elliptic equations on smooth and bounded two dimensional domains. Then, we are able to generalize to the uniformly elliptic case some sharp estimates for Liouville type equations due to Brezis-Merle [7] and, in the same spirit of [3], a "mass" quantization result due to Y.Y. Li [21]. As a consequence, we obtain uniform a priori estimates for solutions of the corresponding Dirichlet problem. Then, we improve the standard existence theorem derived by direct minimization and, in the same spirit of [17] and [37], obtain the existence of Mountain Pass type solutions