Pointwise Blow-Up Phenomena for a Dirichlet Problem
Pierpaolo Esposito, Maristella Petralla · Communications in Partial Differential Equations · 2011
For the Dirichlet problem with Ω ⊂ ℝ N , N ≥ 2, a bounded domain and p > 1, blow-up phenomena necessarily arise as λ → + ∞. 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.