The number of positive solutions to the Brezis-Nirenberg problem

TL;DR

This paper investigates the exact number and local uniqueness of positive solutions to the Brezis-Nirenberg problem for small perturbations, using blow-up analysis and Pohozaev identities, revealing their dependence on the domain's Green's function.

Contribution

It provides new local uniqueness results and an explicit description of the number of positive solutions based on the Green's function, advancing understanding of the solution structure.

Findings

Established local uniqueness of multi-peak solutions.

Connected solution profiles with the domain's Green's function.

Described the exact number of solutions depending on domain properties.

Abstract

In this paper we are concerned with the well-known Brezis-Nirenberg problem \begin{equation*} \begin{cases} -\Delta u= u^{\frac{N+2}{N-2}}+\varepsilon u, &{\text{in}~\Omega},\\ u>0, &{\text{in}~\Omega},\\ u=0, &{\text{on}~\partial \Omega}. \end{cases} \end{equation*} The existence of multi-peak solutions to the above problem for small $\varepsilon>0$ was obtained by Musso and Pistoia. However, the uniqueness or the exact number of positive solutions to the above problem is still unknown. Here we focus on the local uniqueness of multi-peak solutions and the exact number of positive solutions to the above problem for small $\varepsilon>0$. By using various local Pohozaev identities and blow-up analysis, we first detect the relationship between the profile of the blow-up solutions and the Green's function of the domain $\Omega$ and then obtain a type of local uniqueness results of blow-up solutions. At last we give a description of the number of positive solutions for small positive $\varepsilon$, which depends also on the Green's function.