LAUSR

In this paper, we consider the following elliptic problem \begin{document}$ -\texttt{div}(|\nabla u|^{N-2}\nabla u)+V(x)|u|^{N-2}u = \frac{f(x, u)}{|x|^{\eta}}\; \; \operatorname{in}\; \; \mathbb{R}^{N} $\end{document} and its perturbation problem, where \begin{document}$ N\geq 2 $\end{document} , \begin{document}$ 0 , \begin{document}$ V(x) \geq V_{0 }> 0 $\end{document} and \begin{document}$ f(x, t) $\end{document} has a critical exponential growth behavior. By using the variational technique and the indirection method, the existence of a positive ground state solution is proved. For the perturbation problem, the existence of two distinct nontrivial weak solutions is proved.