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

TL;DR

This paper proves the existence of multi-peak solutions for a slightly supercritical nonlinear PDE in a three-dimensional ball, with peaks approaching the boundary as the supercritical parameter tends to zero.

Contribution

It establishes the existence of positive multi-peak solutions with boundary-approaching peaks for a supercritical PDE in 3D, extending previous work to the case of dimension three.

Findings

Solutions have multiple peaks near the boundary.

Number of peaks varies with the supercritical parameter.

Peaks approach the boundary as the parameter tends to zero.

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}.