Critical groups computations on a class of Sobolev banach spaces via morse index

By Silvia Cingolani, Giuseppina Vannella. 2003

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Qualitative Phenomena for Some Classes of Quasilinear Elliptic Equations with Multiple Resonance (2014, methodology): “W 1,p 0 (Ω) = V ⊕ W and 〈 φ′′(u0)y, y 〉 ≥ β‖y‖(2)b for all y ∈ W, for some β > 0 (see Cingolani & Vannella [15]).”
Nontrivial Solutions for p-Laplace Equations with Right-Hand Side Having p-Linear Growth at Infinity (2005, background): “For this class of approximating functionals, it is possible to prove an extension of the Morse lemma and to establish again a connection between the critical groups and the Morse index (see Cingolani and Vannella, 2003a,b).”