Nondegeneracy of the solutions for elliptic problem with critical exponent

Abstract

This paper deals with the following nonlinear elliptic equation: $$ -\Delta u=Q(|y'|,y'')u^{\frac{N+2}{N-2}},\,\,u>0,\,\,\text{in}\,{ \mathbb{R}}^{N},\,\,u\in D^{1,2}({\mathbb{R}}^{N}), $$ where $(y',y'')\in {\mathbb{R}}^{2}\times {\mathbb{R}}^{N-2}$ , $N\geq 5$ , $Q(|y'|,y'')$ is a bounded nonnegative function in $\mathbb{R}^{2}\times {\mathbb{R}}^{N-2}$ . By using the local Pohozaev identities we prove a nondegeneracy result for the positive solutions constructed in (Peng et al. in J. Differ. Equ. 267:2503–2530, 2019).