Uniqueness of positive solutions of a $n$-Laplace equation in a ball in ${\mathbb{r}}^n$ with exponential nonlinearity

Let $n\ge 2$ and $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a bounded domain. Then by Trudinger-Moser embedding, $W_0^{1,n}(\mathrm{\Omega })$ is embedded in an Orlicz space consisting of exponential functions. Consider the corresponding semi linear $n$-Laplace equation with critical or sub-critical exponential nonlinearity in a ball $B(R)$ with dirichlet boundary condition. In this paper, we prove that under suitable growth conditions on the nonlinearity, there exists an $\gamma_0

0$,andacorresponding$R_0(\gamma_0 ) > 0$suchthatforall$0 < R < R_0$,theproblemadmitsauniquenondegeneratepositiveradialsolution$u$with$|u|_{\infty}\geq \gamma_0$.