Nondegeneracy of the solutions for elliptic problem with critical exponent

Abstract

Abstract This paper deals with the following nonlinear elliptic equation: − Δ u = Q ( | y ′ | , y ″ ) u N + 2 N − 2 , u > 0 , in R N , u ∈ D 1 , 2 ( 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 ′ , y ″ ) ∈ R 2 × R N − 2 $(y',y'')\in {\mathbb{R}}^{2}\times {\mathbb{R}}^{N-2}$ , N ≥ 5 $N\geq 5$ , Q ( | y ′ | , y ″ ) $Q(|y'|,y'')$ is a bounded nonnegative function in R 2 × 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).

Keywords

• Nondegeneracy
• Multibubbling solutions
• Pohozaev identity