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

12 Alves and El HamidiLemma 2.6 If {u n } ⊂ W 1,pΓ (Ω) is a Palais-Smale Sequence to I λ withu n ⇀ u in W 1,pΓ(Ω), then the set I∗ = {i; x i ∈ Ω} ⊂ I given in Lemma2.5 is finite or empty and for some subsequence∇u n (x) → ∇u(x) a.e. in Ω.Now, we establish that the Euler functional I λsatisfies the Palais-Smalecondition under some condition on the level of Palais-Smale sequences.Lemma 2.7 There exists a constant K depending only on p, q, N andΩ such that for every λ > 0, the functional I λ satisfies the Palais-Smale1condition in the interval (−∞, S N pp2N − Kλ ∗p ∗ −q ).Proof. Let {u n } ⊂ W 1,pΓ (Ω) be a Palais-Smale sequence for I λ. Usingstandard arguments it follows that the sequence {u n } is bounded. Thus,from the above lemmas there exists a subsequence still denoted by {u n }and a function u ∈ W 1,pΓ(Ω) such that u n ⇀ u. Using the same argumentsexplored in Alves [3], there is a constant K depending only on p, q, N andΩ such thatI λ (u) ≥ −Kλ p∗p ∗ −q .Let v n = u n − u. Then by Brézis & Lieb [5], we haveand by Sobolev embedding∫The above limits imply‖v n ‖ p = ‖u n ‖ p − ‖u‖ p + o n (1),‖v n ‖ p∗p ∗ = ‖u n‖ p∗p ∗ − ‖u‖p∗ p ∗ + o n(1),Ω∫|u n | q dx →Ω|u| q dx.‖v n ‖ p − ‖v n ‖ p∗p ∗ = o n(1)