Concentration of solutions for a fourth order elliptic equation in $\mathbb{R}^N$

Abstract: In this paper, we study the following fourth order elliptic problem $$ \Delta^2 u=(1+\epsilon K(x)) u^{2^*-1}, \quad x\in \mathbb{R}^N $$ where $2^*=\frac{2N}{N-4}$,$N\geq5$, $ \epsilon>0$. We prove that the existence of two peaks solutions for the above problem, if $K(x)$ has two critical points satisfying certain conditions, provided $\epsilon$ is small enough.