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

14 Alves and El HamidiLemma 2.8 Let β :=for λ ∈ (0, λ ∗ 2), we haveIn particular,p∗p ∗ −q1,p. There exist v ∈ WΓ(Ω) and λ∗ 2 > 0 such thatsup I λ (tv) < 1t≥0 2N S N p − Kλ β .α 2 (λ) < 12N S N p − Kλ β ,where K is the constant found in Lemma 2.7.Proof. Let us denote by {w ε } the family of functions given by( )w ε (x) = C N ε N−pp 2 ε + |x| pp−Npp−1which attains the best constant S of the Sobolev embeddingD 1,p (IR N ) ↩→ L p∗ (IR N ).Without loss of generality, we can consider that 0 ∈ Σ. Moreover, the set∂Ω satisfies the following property (see more details in Adimurthi, Pacellaand Yadava [1]):There exist δ > 0, an open neighborhood V of 0 and a diffeomorphismΨ : B δ (0) −→ V which has a jacobian determinant equal to one at 0, withΨ(B + δ ) = V ∩ Ω, where B+ δ = B δ(0) ∩ {x ∈ R N : x N > 0}.Let φ ∈ C ∞ 0 (R N ) such that φ(x) = 1 in a neighborhood of the origin.We define u ε (x) = φ(x)w ε (x). Taking v ε = u ε‖u ε‖ pand using the same∗type of arguments developed in Medeiros [15], we get the following estimates(see Tarantello [16] and Wang [18] for the case p = 2)⎧⎪⎨‖∇v ε ‖ p p =⎪⎩S2 p NS2 p N− Cε p−1p+ o(ε p−1p ) + O()− Cε N−ppf(ε) + O(ε N−ppε N−pp)if N ≥ p 2if N < p 2