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

Abstract

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 $\mathbb{R}^3$, $K(x)$ is a nonnegative radial function under suitable condition 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 \cite{LiuPeng2016}.