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*} -Δu=K(x) u^{5+μ},\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 $\mu \to 0$. Moreover, the number of peaks varies with the parameter $\mu$ as $\mu$ goes to $0^{+}$. Note that the case $n\ge 4$ was considered by Liu and Peng \cite{LiuPeng2016}.