Symmetries and blow-up phenomena for a Dirichlet problem with a large parameter

For the Dirichlet problem $-\Delta u+\lambda V(x) u=u^p$ in $\Omega \subset \mathbb R^N$, $N\geq 3$, in the regime $\lambda \to +\infty$ we aim to give a description of the blow-up mechanism. For solutions with symmetries an uniform bound on the ``invariant" Morse index provides a localization of the blow-up orbits in terms of c.p.'s of a suitable modified potential. The main difficulty here is related to the presence of fixed points for the underlying group action.