We study the following semilinear elliptic equations \(\begin{aligned} -\Delta u+V(x)u=u^{\frac{N+2}{N-2}-\epsilon },\;\; u>0\;\;\text {in}\;\mathbb {R}^N, \end{aligned}\) where V(x) is a bounded nonnegative \(C^1\) function, \(\epsilon >0\) , \(N=5,6\) . If V(x) has k stable critical points, then we show the existence of multi-peak solutions to the above equations in lower dimensions by combining a finite reduction argument and local Pohozaev type identities. The existence of multi-peak solutions in higher dimensions \(N\ge 7\) has been obtained by Hirano et al. (Comm Pure Appl Anal 4:143–164, 2005). We believe that the new ideas and methods developed in the present paper can be employed to study many other nonlocal and higher order elliptic equations with critical nonlinearity.
