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

Abstract

Let $n\ge 2$ 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 $\Vert u{\Vert }_{\infty }\ge {\gamma }_0$.