A uniqueness result for a semilinear elliptic equation in symmetric domains

Abstract

We prove that the problem $$ \begin{cases} -\Delta u=u^p\quad & \text{in $\Omega$}\\ u>0\quad & \text{in $\Omega$ } \\ u=0\quad & \text{on $\partial\Omega$} \end{cases} $$ has only one solution if $\Omega$ is a convex symmetric domain of $\Bbb R^N $, $N\ge3$ and $p <{{N+2}\over{N-2}}$ is close to ${{N+2}\over{N-2}}$. Moreover, we show that this solution is nondegenerate.