On the uniqueness of solutions for a semilinear elliptic problem in convex domains

Abstract

We exhibit a class of convex and nonsymmetric domains $\Omega$ in $\mathbb R^N,$ $N\ge4,$ such that the slightly subcritical problem $$ \begin{cases} -\Delta u=u^{{N+2\over N-2}-{\varepsilon}} & \text{ in $\Omega,$ } \\ u>0 & \text{ in $\Omega,$ } \\ u=0 & \text{ on $\partial\Omega $} \end{cases} $$ does not have any solutions blowing up at more than one point in $\Omega$ as ${\varepsilon}$ goes to zero. Moreover if $\Omega$ is a small perturbation of a convex and symmetric domain, we prove that such a problem has a unique solution provided ${\varepsilon}$ is small enough.