local uniqueness of the elliptic problem with critical exponent via Pohozaev identities

Abstract

We consider the following elliptic equations with critical exponents:\begin{align*}-\Delta u=Q(|y|)u^{2^*-1},\;\;\; u>0\;\;\;\hbox{in } \mathbb{R}^N,\end{align*}where $2^*=\frac{2N}{N-2}$, $N\geq7, Q(|y|)$ is a positive bounded smooth function.Using the Lyapunov-Schmidt reduction method, \cite{WY} shows the equation has infinitely many bubbling solutions. In this paper, Combining the reduction method and the local Pohozaev identities, we will provethe bubbling solutions obtained in \cite{WY} are local uniqueness. AMS Subject Classification : 35B40, 35B45, 35J40