RESEARCH
The number of positive solutions to the Brezis-Nirenberg problem

2020-11-19　10:00 — 11:00

In this talk, 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 above problem for small $\varepsilon>0$ was obtained in [Musso-Pistoia 2002]. However, the uniqueness or the exact  number of positive solutions to above problem is still unknown. Here we focus on the local uniqueness of  multi-peak solutions and  the exact  number of positive solutions to 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.

This is a work jointed with Daomin Cao and Shuangjie Peng.