We study positive solutions $u$ to $\Delta u+f(u)=0$ in $\Omega$, $u=0$ on $\partial\Omega$, and we address the following question: if $\Omega$ is a small perturbation of a ball, is $u$ a small perturbation of a radially symmetric function? We prove two theorems which give an affirmative answer under different assumptions on the non-linearity $f$ and on the topologies in which perturbations are considered.
