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Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates
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
 and, in the same spirit of , a "mass"
quantization result due to Y.Y. Li . 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  and , obtain the existence of
Mountain Pass type solutions.