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

TL;DR

This paper proves the uniqueness of positive radial solutions for a semi-linear $n$-Laplace equation with exponential nonlinearity in small balls, under certain growth conditions, extending understanding of critical exponential problems.

Contribution

It establishes conditions for the existence and uniqueness of positive solutions to an $n$-Laplace equation with exponential nonlinearity in small domains.

Findings

Unique positive radial solutions exist for small radii.

Solutions have a uniform positive lower bound.

The results depend on growth conditions of the nonlinearity.

Abstract

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 $\|u\|_{\infty}\geq \gamma_0$.