On a quasilinear mean field equation with exponential nonlinearity

TL;DR

This paper studies a mean field equation with exponential nonlinearity involving the N-Laplace operator, establishing solution compactness and existence results in non-resonant regimes through asymptotic and variational methods.

Contribution

It provides a detailed asymptotic analysis and proves solution compactness and existence for a class of mean field equations with exponential nonlinearities.

Findings

Quantization property in the non-compact case

Solution set is compact in the non-resonant regime

Existence of solutions established via variational methods

Abstract

The mean field equation involving the $N$-Laplace operator and an exponential nonlinearity is considered in dimension $N\geq2$ on bounded domains with homogenoeus Dirichlet boundary condition. By a detailed asymptotic analysis we derive a quantization property in the non-compact case, yielding to the compactness of the solutions set in the so-called non-resonant regime. In such a regime, an existence result is then provided by a variational approach.