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

Nehari manifold and existence of positive solutions 11Remark 2.2 Until now, the minimizing sequences we consider are notnonnegative. Notice that for every u ∈ W 1,pΓ(Ω) \ {0} and 0 < λ < λ∗ 1, onehas Ĩλ(t, |u|) = Ĩλ(t, u), t 1 (|u|, λ) = t 1 (u, λ) and t 2 (|u|, λ) = t 2 (u, λ). Thus,every minimizing sequence (u n ) ⊂ S of (2.10) or (2.11) can be considered asa sequence of nonnegative functions.Hereafter, we will assume the sequences U n and V n , defined in Lemma 2.3,to be nonnegative.Since we consider mixed Dirichlet-Neumann boundary conditions inProblem (1.1), we will need the following estimate, due to Cherrier [7]:Lemma 2.4 For each τ > 0, there exists M τ > 0 such that[ S]− τ ‖u‖ p p ≤ ∗ ‖∇u‖p p + M τ ‖u‖ p p, ∀u ∈ W 1,p (Ω).2 p NAt this stage, we will state a version of the Concentration CompactnessLemma due P. L. Lions [13, 14], which follows using similar argumentsexplored in the case W 1,p0 (Ω) together with the Cherrier’s inequality. Inthe W 1,p (Ω) case, we can refer the reader to [15] by Medeiros.Lemma 2.5 Let {u n } be a weakly convergent sequence in W 1,p (Ω) with weaklimit u, and such that:i) ‖∇u n ‖ p p → µ weakly-* in the sense of measure,ii) ‖u n ‖ p∗p∗ → ν weakly-* in the sense of measure.Then, for some finite index set I we have:⎧1) ν = ‖u‖ p∗p⎪⎨+ ∑ ∗ j∈I ν jδ xj , ν j > 0,2) µ ≥ ‖∇u‖ p p + ∑ j∈I µ jδ xj , µ j > 0,⎪⎩p3) if x j ∈ Ω then Sν∗j ≤ µ j ,4) if x j ∈ Σ then Spν2 p p ∗j ≤ µ j .NpΓFinally, adapting well know arguments found in [2], [11], and the previouslemma, we can prove the following lemma.