$N$-Laplacian equations in $\mathbb{R}^N$ with critical growth

Abstract

We study the existence of nontrivial solutions to the following problem: $$\{\begin{array}{l}u\in W^{1,N}\left({ℝ}^N\right),u\ge 0\text{and}\\ -div\left({\left|\nabla u\right|}^{N-2}\nabla u\right)+a\left(x\right){\left|u\right|}^{N-2}u=f\left(x,u\right)\text{in}{ℝ}^N\left(N\ge 2\right),\end{array}$$ where $a$ is a continuous function which is coercive, i.e., $a\left(x\right)\to \infty \text{as}\left|x\right|\to \infty$ and the nonlinearity $f$ behaves like $exp\left(\alpha {\left|u\right|}^{N/\left(N-1\right)}\right)$ when $\left|u\right|\to \infty$.