Existence of positive solutions for a class of almost critical problems on an annulus

TL;DR

This paper investigates the existence and behavior of multi-peak positive solutions for almost critical elliptic problems on an annulus, revealing how the annulus's geometry influences solution concentration as parameters vary.

Contribution

It provides explicit analysis of solution structures on an annulus using Green and Robin functions, highlighting geometric effects in near-critical elliptic problems.

Findings

In the subcritical case, the annulus becomes thinner with more peaks.

In the supercritical case, the hole of the annulus becomes very small.

Explicit use of Green and Robin functions to analyze solution behavior.

Abstract

In this paper we will consider multi-peaks positive solutions for a class of slightly subcritical or slightly supercritical elliptic problems on an annulus with Dirichlet boundary conditions. By using the explicit form of the Green function and of the Robin function on the annulus, we prove that the annulus becomes thinner and thinner when the number of bumps increases for the slightly subcritical case, while the hole of the annulus is very small for the slightly supercritical case.