Semilinear Dirichlet problems for the $N$-Laplacian in $\mathbb{R}^N$ with nonlinearities in the critical growth range

Abstract

The goal of this paper is to study the existence of solutions for the following class of problems for the $N$-Laplacian $$ u\in W_0^{1,N}(\Omega),\quad u\geq 0\ \ \text{ and } \ \ -\Delta_Nu\equiv -\text{div}( | \nabla u | ^{N-2}\nabla u)=f(x,u) \,\,\, \rm{in}\,\,\, \Omega , $$ where $\Omega $ is a bounded smooth domain in $\Bbb R^N$ with $N\geq 2$ and the nonlinearity $f(x,u)$ behaves like $\exp (\alpha | u | ^{\frac N{N-1}})$ when $ | u | \to \infty .$