## Differential and Integral Equations

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

João Marcos B. do Ó

#### 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 $| u | \to \infty .$

#### Article information

Source
Differential Integral Equations, Volume 9, Number 5 (1996), 967-979.

Dates
First available in Project Euclid: 6 May 2013

Marcos B. do Ó, João. Semilinear Dirichlet problems for the $N$-Laplacian in $\mathbb{R}^N$ with nonlinearities in the critical growth range. Differential Integral Equations 9 (1996), no. 5, 967--979. https://projecteuclid.org/euclid.die/1367871526