Nehari manifold and existence of positive solutions to a class of ...

Nehari manifold and existence of positive solutions 13and1p ‖v n‖ p − 1 p ∗ ‖v n‖ p∗p ∗ = c − I λ(u) + o n (1).Since the sequence (v n ) n is bounded in W 1,p (Ω), there exist l ≥ 0 and asubsequence, still denote by {v n }, verifyingΓ‖v n ‖ p → l.Hence,‖v n ‖ p∗p ∗ → l.Using Cherrier’s inequality and passing to the limit n → ∞, we obtain[ S2 p N− τ]l pp ∗ ≤ l ∀τ > 0,that is,S2 p Nl pp ∗ ≤ l.Now, we claim that l = 0. Indeed in one hand, if l > 0 the last inequalityimpliesOn the other hand,and thenl ≥ S N p2 .1N l = c − I λ(u),c ≥ 12N S N p− Kλp ∗p ∗ −q ,which contradicts the hypothesis. Therefore, l = 0 and we conclude thatu n → u in W 1,pΓ (Ω). □