Solutions to a perturbed critical semilinear equation concerning the $N$-Laplacian in $\Bbb R^{N}$

summary:The aim of this paper is to study the existence of variational solutions to a nonhomogeneous elliptic equation involving the $N$-Laplacian $$ - \Delta_N u \equiv - \operatorname{div} (|\nabla u|^{N-2} \nabla u) = e(x,u) + h(x) \text{ in } \Omega $$ where $u \in W_0^{1,N}(\Bbb R^{N})$, $\Omega$ is a bounded smooth domain in $\Bbb R^{N}$, $N \geq 2$, $e(x,u)$ is a critical nonlinearity in the sense of the Trudinger-Moser inequality and $h(x) \in (W_0^{1,N})^*$ is a small perturbation