Nondegeneracy of the solutions for elliptic problem with critical exponent

This paper deals with the following nonlinear elliptic equation: $-\mathrm{\Delta }u=Q(|y^{\prime }|,y^{″})u^{\frac{N+2}{N-2}},u>0,\text{in}{\mathbb{R}}^N,u\in D^{1,2}({\mathbb{R}}^N),$ \( -\Delta u=Q(|y'|,y'')u^{\frac{N+2}{N-2}},\,\,u>0,\,\,\text{in}\,{ \mathbb{R}}^{N},\,\,u\in D^{1,2}({\mathbb{R}}^{N}), \) where $(y^{\prime },y^{″})\in {\mathbb{R}}^2\times {\mathbb{R}}^{N-2}$ $(y',y'')\in {\mathbb{R}}^{2}\times {\mathbb{R}}^{N-2}$ , $N\ge 5$ $N\geq 5$ , $Q(|y^{\prime }|,y^{″})$ $Q(|y'|,y'')$ is a bounded nonnegative function in ${\mathbb{R}}^2\times {\mathbb{R}}^{N-2}$ $\mathbb{R}^{2}\times {\mathbb{R}}^{N-2}$ . 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).