Wen Yang

Analytic aspects of the Tzitzéica equation: blow-up analysis and existence results

We are concerned with the following class of equations with exponential nonlinearities:

Non degeneracy, Mean Field Equations and the Onsager theory of 2D turbulence

begin{aligned} Delta u+h_1e^u-h_2e^{-2u}=0 qquad mathrm {in}~B_1subset mathbb {R}^2, end{aligned}

Sharp estimates for solutions of mean field equations with collapsing singularity

Δu+h1eu-h2e-2u=0inB1⊂R2,which is related to the Tzitzéica equation. Here

Classification of blow-up limits for the sinh-Gordon equation

h_1, h_2

On large ring solutions for Gierer–Meinhardt system in R3

h1,h2 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.

On the Topological degree of the Mean field equation with two parameters

For all rank two Toda systems with an arbitrary singular source, we use a unified approach to prove: (i) The pair of local masses