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

Let $n \geq 2$ and $\Omega \subset \mathbb{R}^n$ be a bounded domain. Then by Trudinger-Moser embedding, $W_0^{1,n}(\Omega)$ 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 $\gamma_0 > 0$, and a corresponding $R_0(\gamma_0 ) > 0$ such that for all $0 < R < R_0$ , the problem admits a unique non degenerate positive radial solution $u$ with $\|u\|_{\infty}\geq \gamma_0$.