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**1 - 1**of**1**### SYMMETRIES AND BLOW-UP PHENOMENA FOR A DIRICHLET PROBLEM WITH A LARGE PARAMETER

, 2012

"... For the Dirichlet problem −∆u+λV (x)u = up in Ω ⊂ RN, N ≥ 3, in the regime λ → + ∞ 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 mo ..."

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For the Dirichlet problem −∆u+λV (x)u = up in Ω ⊂ RN, N ≥ 3, in the regime λ → + ∞ 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.