Positive solutions of slightly supercritical elliptic equations in symmetric domains

This paper deals with existence and multiplicity of solutions for problem $P(\e,\Omega)$ below, which concentrate and blow-up at a finite number of points as $\e\to 0$. We give sufficient conditions on $\Omega$ which guarantee that the following property holds: there exists $\bar k (\Omega)$ such that, for each $k\geq \bar k (\Omega)$, problem $P(\e,\Omega)$, for $\e>0$ small enough, has at least one solution blowing up as $\e\to 0$ at exactly $k$ points. Exploiting the properties of the Green and Robin functions, we also prove that the blow up points approach the boundary of $\Omega$ as $k\to\infty$. Moreover we present some examples which show that $P(\e,\Omega)$ may have $k-$spike solutions of this type also when $\Omega$ is a contractible domain, not necessarily close to domains with nontrivial topology and, for $\e>0$ small and $k$ large enough, even when it is very close to star-shaped domains.