Solutions with large number of peaks for a slightly supercritical nonlinear equation in dimension three

We investigate the existence of solutions to the semilinear equation with a slightly supercritical exponent in dimension three,$ \begin{align*} -\Delta u = K(x) u^{5+\mu},\quad u>0 \; \text{in}\; \mathbf{B}, \quad u = 0 \; \text{on}\; \partial \mathbf{B}, \end{align*} $where $ \mu >0 $, $ \mathbf{B} $ is the unit ball in $ \mathbb{R}^3 $, and $ K(x) $ is a nonnegative radial function under suitable conditions on $ K $. We prove the existence of positive multi-peak solutions for $ \mu>0 $ small enough. All peaks of our solutions approach the boundary $ \partial\mathbf{B} $ as $ \mu\rightarrow 0 $. Moreover, the number of peaks varies with the parameter $ \mu $ as $ \mu $ goes to $ 0^+ $. Note that the case $ n\geq 4 $ was considered by Liu and Peng [16].