Uniqueness of positive solutions of a n -Laplace equation in a ball in RnRn with exponential nonlinearity

Let n≥2n≥2 and Ω⊂RnΩ⊂Rn be a bounded domain. Then by Trudinger–Moser embedding, W01,n(Ω) is embedded in an Orlicz space consisting of exponential functions. Consider the corresponding semilinear n  -Laplace equation with critical or sub-critical exponential nonlinearity in a ball B(R)B(R) with dirichlet boundary condition. In this paper, we prove that under suitable growth conditions on the nonlinearity, there exists an γ0>0γ0>0, and a corresponding R0(γ0)>0R0(γ0)>0 such that for all 0