Pointwise blow-up phenomena for a Dirichlet problem

For the Dirichlet problem $$-\Delta u+\lambda V(x) u=u^p in \Omega\\ u=0 on \partial \Omega,$$ with $\Omega \subset R^N$, $N\geq 2$, a bounded domain and $p>1$, blow-up phenomena necessarily arise as $\lambda \to +\infty$. In the present paper, we address the asymptotic description for pointwise blow-up, as it occurs when either the ``energy" or the Morse index is uniformly bounded. A posteriori, we obtain an equivalence between the two quantities in the form of a double-side bound with essentially optimal constants, a sort of improved Rozenblyum-Lieb-Cwikel inequality for the equation under exam. Moreover, we prove the nondegeneracy of any ``low energy" or Morse index $1$ solution under a suitable condition on the potential.