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We are concerned with the following class of equations with exponential nonlinearities: which is related to the Tzitzéica equation. Here $$h_1, h_2$$ are two smooth positive functions. The purpose of the paper is to initiate the analytical study of the above equation and to give a quite complete picture both for what concerns the blow-up phenomena and the existence issue. In the first part of the paper we provide a quantization of local blow-up masses associated to a blowing-up sequence of solutions. Next we exclude the presence of blow-up points on the boundary under the Dirichlet boundary conditions. In the second part of the paper we consider the Tzitzéica equation on compact surfaces: we start by proving a sharp Moser-Trudinger inequality related to this problem. Finally, we give a general existence result. [ABSTRACT FROM AUTHOR]

We obtain a priori estimates for solutions to the prescribed scalar curvature equation on S 3. The usual non-degeneracy assumption on the curvature function is replaced by a new condition, which is necessary and sufficient for the existence of a priori estimates, when the curvature function is a positive Morse function. [ABSTRACT FROM AUTHOR]

The classical proofs of the De Giorgi–Nash–Moser Theorem are based on the iteration of some inequality through countably many concentric balls. In this note, we present a new approach to the Hölder continuity of solutions to elliptic equations in divergence form, which avoids any form of discrete iteration. In particular, we prove that a suitable energy function satisfies a differential inequality, whose integration yields a new proof of the crucial step in the regularity result. [ABSTRACT FROM AUTHOR]