Multispike Solutions for a slightly subcritical elliptic problem with non-power nonlinearity

TL;DR

This paper constructs multi-peak solutions for a slightly subcritical elliptic problem with a non-power nonlinearity, extending previous results on single-peak solutions to multiple concentration points.

Contribution

It introduces new multi-peak solutions for the elliptic equation with non-power nonlinearity, including sign-changing solutions, for small epsilon.

Findings

Existence of multi-peak solutions at multiple points

Construction of sign-changing solutions

Extension of single-peak results to multiple concentrations

Abstract

In this paper, we are concerned with the following elliptic equation $$\left\{\begin{array}{rrl}-\Delta u&=& |u|^{4/(n-2)}u/[\ln (e+|u|)]^\varepsilon\hbox{ in } \Omega,\\ u&=&0 \hbox{ on }\partial \Omega, \end{array} \right.$$ where $\Omega$ is a smooth bounded open domain in $\mathbb{R}^n, \ n\geq 3$ and $\varepsilon>0$. Clapp et al. in Journal of Diff. Eq. (Vol 275) proved that there exists a single-peak positive solution for small $\varepsilon$ if $n \geq 4$. Here we construct positive as well as changing sign solutions concentrated at several points at the same time.