| Abstract: |
| This talk concerns the uniqueness of the solution of the semilinear elliptic problem
$$
\left\{
\begin{array}{lr}
-\Delta u= u^p & \mbox{ in }\Omega\
u=0 &\mbox{ on }\partial \Omega
\
u>0 & \mbox{ in }\Omega
\end{array}
\right.
$$
when $\Omega\subset \R^2$ is a convex smooth bounded domain and $p\in (1, +\infty)$. We give a partial answer to this longstanding open problem, proving the uniqueness for any finite energy solution when $p$ is sufficiently large, where how large depends on the energy level considered. |
|