On a quasilinear mean field equation with exponential nonlinearity

The mean field equation involving the $N$-Laplace operator and an exponential nonlinearity is considered in dimension $N\ge 2$ 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.