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

Nehari manifold and existence of positive solutions to aclass of quasilinear problemsC.O. Alves a, ∗ and A. El Hamidi ba Departamento de Matemática e Estatistica, UFCG,CEP: 58109-970 , C. Grande -PB, Brazilb Laboratoire de Math. & Applications,Université de la Rochelle,17042 La Rochelle, France.AbstractIn this paper, existence and multiplicity results to the following nonlinearelliptic equation−∆ p u = λ|u| q−2 u + |u| p∗ −2 u, u > 0 in Ω ⊂ R N ,together with mixed Dirichlet-Neumann or Neumann boundary conditions,are established. Here, ∆ p u denotes the p-Laplacian operator, 1 < q < p < N,p ∗ =NpN−pand λ is a positive real parameter. The study is based on theextraction of Palais-Smale sequences in the Nehari manifold.MSC: 35A15; 35B33; 35H39Keywords: Variational Method, Quasilinear problem, Critical Sobolev exponent∗ Research Supported by Millennium Institute for the Global Advancement of BrazilianMathematics - IM-AGIMB1

2 Alves and El Hamidi1 IntroductionIn this paper, we deal with the existence of multiple solutions to the boundaryvalue problems⎧⎨⎩⎧⎨⎩−∆ p u = λ|u| q−2 u + |u| p∗−2 u in Ω,u = 0 on Γ,(1.1)∂u∂ν on Σ,−∆ p u = λ|u| q−2 u + |u| p∗ −2 u in Ω,p−2 ∂u|∇u| = ∂ν −a(x)|u|p−2 uon ∂Ω,(1.2)with respect to the real parameter λ. Here, Ω is a bounded domain in R Nwith smooth boundary ∂Ω = Γ∪Σ, where Γ, Σ are smooth (N-1)-dimensionalsubmanifolds of ∂Ω with positive measures such that Γ ∩ Σ = ∅, ∆ p is the∂p-Laplacian, is the outer normal derivative, and ∂νp∗ = Np . ThroughoutN−pthis paper, the function a is assumed to be in L ∞ (∂Ω), a(s) ≥ a 0 > 0 almosteverywhere on a subset of ∂Ω with positive measure.In a recent paper, K. J. Brown & Y. Zhang [6] have studied a subcriticalsemilinear elliptic equation with a sign-changing weight function and abifurcation real parameter in the case p = 2 and Dirichlet boundaryconditions.Exploiting the relationship between the Nehari manifold andfibering maps (i.e., maps of the form t ↦→ J λ (tu) where J λ is the Euler functionassociated with the equation), they gave an interesting explanation of a wellknown bifurcation result. In fact, the nature of the Nehari manifold changesas the parameter λ crosses the bifurcation value. In Tarantello [16], usingthe same type of approach the critical case has been studied also assumingp = 2, q = 2 and Neumann boundary conditions.In this work, we exploit similar facts to show the existence of multiplenontrivial positive solutions to (1.1) and (1.2). The idea of our approach canbe summarized as follows: Let I λ (resp. J λ ) the Euler functional associated toProblem (1.1) (resp. Problem (1.2)) defined on W 1,pΓ(Ω) (resp. on W 1,p (Ω)),

Nehari manifold and existence of positive solutions 3whereW 1,pΓ (Ω) = {u ∈ W 1,p (Ω); u| Γ = 0}is the closure of C 1 0(Ω ∪ Γ) with respect to the norm of W 1,p (Ω) (we refer thereader to the paper by Colorado & Peral [8] for a complete study in the casep = 2).For each u ∈ W 1,pΓ(Ω) \ {0} (resp. W 1,p (Ω) \ {0}) λ > 0, we determine thereal values of t (in terms of u and λ) such that tu belongs to the Neharimanifold:N Iλ = { v ∈ W 1,pΓ (Ω) \ {0} : I′ λ(v)(v) = 0 }(resp. N Jλ = { v ∈ W 1,p (Ω) \ {0} : J ′ λ(v)(v) = 0 } ).Then, the variable t is substituted by these special values to obtain newEuler functionals defined on the Nehari manifold. On shows easily that thesefunctionals are bounded below, which allows to find possible critical points byminimization. Moreover, this approach allows the simultaneous constructionof Palais-Smale sequences, in the Nehari manifold, giving directly existenceand multiplicity results [10].Let us mention that in the case p = 2 and with a subcritical concaveconvexnonlinearity, this problem was studied recently by Colorado & Peral[8]. The authors showed that there is a special value Λ of the parameter λsuch that Problem (1.1) admits at least two positive solutions for λ ∈ (0, Λ),admits at least one positive solution for λ = Λ and admits no positive solutionfor λ > Λ.In our opinion, an interesting question which is not treatedin our paper, is to characterize Λ as a bifurcation value via the dynamicof the Nehari manifold with respect to the parameter λ (see [6]).Someresults involving the p-Laplacian operator, concave-convex nonlinearity andDirichlet boundary conditions can be found in the papers of Garcia Azorero& Peral Alonso [11] and Ambrosetti, Garcia Azorero & Peral Alonso [4].This paper is organized as follows: Section 2 is devoted to Problem (1.1)and Section 3 is the subject of Problem (1.2).

4 Alves and El Hamidi2 The mixed Dirichlet-Neumann Problem(1.1)It is well known that weak solutions of (1.1) correspond to critical points ofthe C 1 functional I λ : W 1,p (Ω) −→ R, given bywhere∫P (u) =ΩΓI λ (u) = 1 p P (u) − λ q Q(u) − 1 p ∗ P ∗ (u),∫|∇u| p dx, Q(u) =Ω∫|u| q dx and P ∗ (u) =Ω|u| p∗ dx.Using the fact that Γ has strictly positive measure, the Poincaré inequality isstill available in the space W 1,p (Ω), hence it can endowed with the followingnormΓ‖u‖ ={ ∫ |∇u| p dxΩ(see some commentaries in Kesavan’s book [12, page 125], for the case p=2).In the sequel, || ||, || || q and || || p ∗ will denote the norms on W 1,pΓ(Ω), Lq (Ω)and L p∗ (Ω) respectively. We introduce the modified functional Ĩλ defined onR × W 1,pΓ (Ω) by Ĩλ(t, u) := I λ (tu), (see [19, 9, 10]). For every u ∈ W 1,pΓ(Ω),∂ t Ĩ λ (., u) (resp. ∂ tt Ĩ λ (., u)) is the first (resp. second) derivative of the realvalued function: t ↦→ Ĩλ(t, u).2.1 Preliminary resultsSince the functional Ĩλ is even in t and that we are interested by the nontrivialsolutions of (1.1), we limit our study for t > 0 and u ∈ W 1,pΓ(Ω) \ {0}.Lemma 2.1 For every u ∈ W 1,p (Ω) \ {0}, there is a unique λ(u) > 0 suchΓ} 1p,that the real valued function t ↦→ ∂ t Ĩ λ (t, u) has exactly two positive zeros(resp. one positive zero) if 0 < λ < λ(u) (resp. λ = λ(u)). This functionhas no zero for λ > λ(u).

Nehari manifold and existence of positive solutions 5Proof. Let u be an arbitrary element of W 1,pΓ(Ω) \ {0} and let us write∂ t Ĩ λ (t, u) = t q−1 H λ (t, u), where H λ (t, u) = t p−q P (u) − λQ(u) − t p∗ −q P ∗ (u).Then∂ tt Ĩ λ (t, u) = (q − 1)t q−2 H λ (t, u) + t q−1 ∂ t H λ (t, u),holds true, with∂ t H λ (t, u) = t p−q−1 { (p − q)P (u) − (p ∗ − q)t p∗ −p P ∗ (u) } .The real valued function H λ (., u) is increasing on (0, t(u)), decreasing on(t(u), +∞) and attains its unique maximum for t = t(u), wheret(u) =( ) 1p − q P (u)p ∗ −p. (2.3)p ∗ − q P ∗ (u)Thus, the function H λ (., u) has two positive zeros (resp. one positive zero) ifH λ (t(u), u) > 0 (resp. if H λ (t(u), u) = 0) and has no zero if H λ (t(u), u) < 0.On the other hand, a direct computation givesH λ (t(u), u) = p∗ − pp − qSimilarly, H λ (t(u), u) > 0 (resp.λ > λ(u)) and H λ(u) (t(u), u) = 0, wherewithλ(u) = Θ( ) p ∗ −qp − q P (u)p ∗ −pP ∗ (u) − λQ(u).p ∗ − q P ∗ (u)Θ = p∗ − pp − qH λ (t(u), u) < 0) if λ < λ(u) (resp.Pp∗ −qp ∗ −p (u), (2.4)Q(u) P ∗ p−qp ∗ −p (u)( ) p ∗ −qp − qp ∗ −p.p ∗ − qIt follows that if λ ∈]0, λ(u)[, the real valued function ∂ t Ĩ λ (., u) has twopositive zeros, denoted by t 1 (u, λ) and t 2 (u, λ), verifying 0 < t 1 (u, λ)

6 Alves and El Hamidit(u) < t 2 (u, λ).Since, H λ (t 1 (u, λ), u) = H λ (t 2 (u, λ), u) = 0, ∂ t H λ (t, u) > 0 for t < t(u) and∂ t H λ (t, u) < 0 for t > t(u), it follows that∂ tt Ĩ λ (t 1 (u, λ), u) > 0 and ∂ tt Ĩ λ (t 2 (u, λ), u) < 0.This means that the real valued function Ĩλ(., u), defined for t > 0, achievesits unique local minimum (resp. unique local maximum) at t = t 1 (u, λ) (resp.t = t 2 (u, λ)).Notice that for every u ∈ W 1,pΓ(Ω) \ {0} and λ ∈ (0, λ(u)), t 1(u, λ)u andt 2 (u, λ)u belong to the Nehari manifold [19] defined by□Now, we introduceN Iλ := {u ∈ W 1,pΓ (Ω) \ {0} : I′ λ(u)u = 0}.λ ∗ 1 := inf { λ(u) : u ∈ W 1,pΓ (Ω) \ {0}} . (2.5)If S Γ,q (resp. S Γ ) denotes the best Sobolev constant of the embeddingΓ (Ω) ⊂ Lq (Ω) (resp. W 1,pΓ(Ω) ⊂ Lp∗ (Ω)), thenW 1,pSince ∂ t Ĩ λ (t 1 (u, λ), u) = 0 (resp.λ ∗ 1 ≥ ĈSq/p Γ,q S NpΓ(1−q/p) > 0.∂ t Ĩ λ (t 2 (u, λ), u) = 0) for every u ∈W 1,pΓ(Ω) \ {0}, it follows that the functional u ↦−→ Ĩλ(t 1 (u, λ), u) (resp.u ↦−→ Ĩλ(t 2 (u, λ), u)) is bounded below on W 1,p (Ω) \ {0}. Thus, for everyλ ∈ (0, λ ∗ 1), we defineΓ}α 1 (λ) = inf{Ĩλ (t 1 (u, λ), u) : u ∈ W 1,pΓ (Ω) \ {0} , (2.6)}α 2 (λ) = inf{Ĩλ (t 2 (u, λ), u) : u ∈ W 1,pΓ (Ω) \ {0} . (2.7)

Nehari manifold and existence of positive solutions 7Remark 2.1 For every real number γ > 0, we have(Ĩ λ γt, u )=γ Ĩλ(t, u),(∂ t Ĩ λ γt, u )= 1 γ γ ∂ tĨλ(t, u),(∂ tt Ĩ λ γt, u )= 1 γ γ ∂ ttĨλ(t, u),2it follows thatt 1 (u, λ) = 1 ( ) uγ t 1γ , λ , (2.8)t 2 (u, λ) = 1 ( ) uγ t 2γ , λ . (2.9)Therefore, α 1 (λ) and α 2 (λ) can be rewritten as followswhere S is the unit sphere of W 1,pΓ(Ω).α 1 (λ) = infu∈S Ẽλ(t 1 (u, λ), u), (2.10)α 2 (λ) = infu∈S Ẽλ(t 2 (u, λ), u), (2.11)Lemma 2.2 Let (u n ) ⊂ S be a minimizing sequence of (2.10) (resp.(2.11)) and U n := t 1 (u n , λ)u n (resp. V n := t 2 (u n , λ)u n ). Thenof(i)lim supn→+∞||U n || < +∞(resp. lim supn→+∞||V n || < +∞),(ii)lim inf ||U n|| > 0n→+∞(resp. lim infn→+∞ ||V n|| > 0).Proof.(i) Let (u n ) ⊂ S be a minimizing sequence of (2.10). Since∂ t Ĩ λ (t 1 (u n , λ), u n ) = 0, it follows that||U n || p = λ||U n || q q + ||U n || p∗p∗. (2.12)

8 Alves and El HamidiSimilarly, since ∂ tt Ĩ λ (t 1 (u n , λ), u n ) > 0, it follows that(p − 1)||U n || p − λ(q − 1)||U n || q q − (p ∗ − 1)||U n || p∗p∗ > 0. (2.13)Combining (2.12) and (2.13), we get I λ (U n ) < 0, for every n.Suppose that there is a subsequence of (U n ), still denoted by (U n ) suchthat lim ||U n|| = +∞. It is well known that there is some constant C pn→+∞ ∗ ,qsuch that ||U n || q ≤ C p∗,q ||U n || p ∗ for every n, thenlim ||U n|| p ∗ = +∞. Usingn→+∞the fact that 0 < q < p ∗ we get ||U n || q q = o n(||U n || p∗p ∗ ), and consequently||U n || p = ||U n || p∗p ∗(1 + o n(1)).Thus,( ) 1I λ (U n ) = ||U n || p∗p ∗ N + o n(1) ,which implies that I λ (U n ) tends to +∞ as n goes to +∞ and this isimpossible. Hence, we conclude that lim sup ||U n || < +∞.n→+∞The same arguments with a minimizing sequence (u n ) of (2.11) show thatlim sup ||V n || < +∞.n→+∞(ii) Let (u n ) ⊂ S be a minimizing sequence of (2.10) and suppose that thereis a subsequence of (U n ), still denoted by (U n ) such that lim ||U n|| = 0.n→+∞It follows that lim I λ(U n ) = 0 i.e. α 1 (λ) = 0, which is impossible sincen→+∞Ĩ λ (t 1 (u n , λ), u n ) < 0 for every n.Let (u n ) ⊂ S be a minimizing sequence of (2.11). Since∂ t Ĩ λ (t 2 (u n , λ), u n ) = 0 and ∂ tt Ĩ λ (t 2 (u n , λ), u n ) < 0 it follows that{ ||Vn || p − λ||V n || q q − ||V n || p∗p ∗ = 0,(p − 1)||V n || p − λ(q − 1)||V n || q q − (p ∗ − 1)||V n || p∗p ∗ < 0.Combining the two last inequalities we obtain, for every n,(p − q)||V n || p < (p ∗ − q)||V n || p∗p ∗ ≤ (p∗ − q)S Γp ∗ /p ||V n || p∗ ,

Nehari manifold and existence of positive solutions 9via the continuous embedding W 1,pΓ (Ω) ⊂ Lp∗ (Ω). Then (p − q) ≤ (p ∗ −pq)S ∗ /p Γ ||V n || p∗−p . Now, suppose that there is a subsequence of (V n ), stilldenoted by (V n ) such thatwhich is impossible.lim ||V n|| = 0. This implies that p − q ≤ 0,n→+∞□Lemma 2.3 Let (u n ) ⊂ S be a minimizing sequence of (2.10) (resp.(2.11)). Then, (U n ) := (t 1 (u n , λ)u n ) (resp. (V n ) := (t 2 (u n , λ)u n )) are Palais-Smale sequences for the functional I λ .Proof.We will show this lemma only for the sequence (U n ), the proof for(V n ) can be done in the same way.First, according to the previous lemma, it is clear that (U n ) is bounded inW 1,pΓ1,p(Ω). On the other hand, notice that for every u ∈ W (Ω) \ {0} andΓofλ ∈ (0, λ ∗ 1), we have ∂ t Ĩ λ (t 1 (u, λ), u) = 0 and ∂ tt Ĩ λ (t 1 (u, λ), u) ≠ 0.Theimplicit function theorem implies that t 1 (u, λ) is C 1 with respect to u sinceĨ is. Let us introduce the C 1 functional I λ defined on S byThenI λ (u) = Ĩλ(t 1 (u, λ), u) ≡ I λ (t 1 (u, λ)u).α 1 (λ) = infu∈S I λ(u) andlim I λ(u n ) = α 1 (λ).n→+∞Using the Ekeland variational principle on the complete manifold (S, || ||) tothe functional I λ , we conclude that|I ′ λ(u n )(ϕ n )| ≤ 1 n ||ϕ n||, for every ϕ n ∈ T un S,where T un S is the tangent space to S at the point u n . Moreover, for everyϕ n ∈ T un S, one hasI ′ λ(u n )(ϕ n ) = ∂ t Ĩ λ (t 1 (u n , λ), u n )t ′ 1(u n , λ)(ϕ n ) + ∂ u Ĩ λ (t 1 (u n , λ), u n )(ϕ n ),= ∂ u Ĩ λ (t 1 (u n , λ), u n )(ϕ n ),

10 Alves and El Hamidisince ∂ t Ĩ λ (t 1 (u n , λ), u n ) ≡ 0, where t ′ 1(u n , λ) denotes the derivative of t 1 (., λ)with respect to its first variable at the point (u n , λ).Furthermore, letπ : W 1,pΓ(Ω) \ {0} −→((0, +∞) ×)Suu ↦−→ ||u||, := (π||u|| 1 (u), π 2 (u)).Applying Hölder’s inequality, we get for every (u, ϕ) ∈ ( W 1,pΓ (Ω) \ {0}) ×W 1,pΓ (Ω): {|π′1 (u)(ϕ)| ≤ ||ϕ||,||π ′ 2(u)(ϕ)|| ≤ 2 ||ϕ||||u|| .From Lemma 2.2, there is a positive constant C such thatt 1 (u n , λ) ≥ C, ∀ n ∈ N.Then for every ϕ ∈ W 1,pΓ(Ω), there are ϕ1 n ∈ R and ϕ 2 n ∈ T un S such that|ϕ 1 n| ≤ ||ϕ||, ||ϕ 2 n|| ≤ 2 ||ϕ|| andCI ′ λ(t 1 (u n , λ)u n )(ϕ) = ∂ t Ĩ λ (t 1 (u n , λ), u n )(ϕ 1 n) + ∂ u Ĩ λ (t 1 (u n , λ), u n )(ϕ 2 n),Therefore,We easily conclude that= ∂ u Ĩ λ (t 1 (u n , λ), u n )(ϕ 2 n),= I ′ λ(u n )(ϕ 2 n).I ′ λ(t 1 (u n , λ)u n )(ϕ) ≤ 1 n ||ϕ2 n||≤ 2nC ||ϕ||.limn→∞ ||I′ λ(U n )|| ∗ = 0,where || || ∗ denotes the norm in the dual space of W 1,pΓ (Ω). □

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.

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)

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). □

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

Nehari manifold and existence of positive solutions 15where C is a positive constant and lim ε→0 f(ε) = +∞. Let δ 2 > 0 be suchthat12N S N p − Kλ β > 0, ∀ λ ∈ (0, δ 2 ).Using the definition of I λ , we getI λ (tv ε ) ≤ tp p ‖v ε‖ p p, ∀ t ≥ 0,which implies that there exists t 0 ∈ (0, 1) satisfyingsup I λ (tv ε ) < 10≤t≤t 02N S N p − Kλ β , ∀ λ ∈ (0, δ 2 ).Analyzing the case N ≥ p 2 , we haveI λ (tv ε ) ≤ 12N S N pp−1− Cεp+ o(ε p−1p ) + O(ε N−pp)− λtqq∫v q ε, ∀ t > 0.Therefore,sup I λ (tv ε ) ≤ 1t≥t 02N S N pp−1− Cεp∫+ o(ε p−1N−pp ) + O(εp) − λtq 0qv q ε.Hence,sup I λ (tv ε ) < 1t≥t 02N S N pp−1− Cεp+ o(ε p−1p ) + O(ε N−pp)− Kλ β , ∀ λ ∈ (0, δ 3 ),whereWe fix ε > 0 such that−Cε p−1pthis is possible since N−ppobtain( tqδ 3 =− p−1p0∫vqε2Kq) 1β−1.+ o(ε p−1p ) + O(≥ (p−1)2pε N−pp)< 0,> 0. If we set λ ∗ 2 = min{δ 2 , δ 3 }, wesup I λ (tv ε ) < 1t≥0 2N S N p − Kλ β , ∀ λ ∈ (0, λ ∗ 2),

16 Alves and El Hamidiand finallyα 2 (λ) < 12N S N p − Kλ β , ∀ λ ∈ (0, λ ∗ 2).The case N < p 2 follows with the same type of arguments.□Theorem 2.1 Let 1 < q < p and ̂λ = min{λ ∗ 1, λ ∗ 2}. Then for λ ∈ (0, ̂λ),problem (1.1) has at least two nonnegative solutions.Proof.Using the above lemmas, there exist two sequences of positivefunctions {U n } and {V n } in W 1,pΓ(Ω) such thatI λ (U n ) → α 1 (λ), ||I ′ λ(U n )|| ∗ → 0 as n → +∞andI λ (V n ) → α 2 (λ),||I ′ λ(V n )|| ∗ → 0 as n → +∞.Notice that for every λ ∈ (0, ̂λ), one hasα 1 (λ) ≤ α 2 (λ) < 12N S N p − Kλ β .Then, there exist two nonnegative functions U λ , V λ ∈ W 1,p (Ω) verifyingandU n −→ U λ in W 1,p (Ω) as n → ∞ΓV n −→ V λ in W 1,pΓ(Ω) as n → ∞.Finally, the solutions {U λ } and {V λ } satisfy the inequalities∂ tt Ĩ λ (1, U λ ) > 0 and ∂ tt Ĩ λ (1, V λ ) < 0,Γwhich imply that U λ ≠ V λ . Finally, applying the Harnack’s inequality (seeTrudinger [17]), we conclude that {U λ } and {V λ } are positive in Ω. Thisachieves the proof. □.

Nehari manifold and existence of positive solutions 173 The Neumann Problem (1.2)In this section, we will state similar results for the Neumann problem (1.2):⎧⎨⎩−∆ p u = λ|u| q−2 u + |u| p∗ −2 u in Ω,p−2 ∂u|∇u| = ∂ν −a(x)|u|p−2 u on ∂Ω.Let us recall that the function a is assumed to be in L ∞ (∂Ω), a(s) ≥ a 0 > 0almost everywhere on on a subset of ∂Ω with positive measure.The Euler functional J λ : W 1,p (Ω) −→ R related to the above problem isgiven byJ λ (u) = 1 p∫Ω|∇u| p + 1 p∫∂Ωa(x)|u| p − λ q∫Ω|u| q − 1 p ∗ ∫Ω|u| p∗ .As in the previous section, for solutions of (1.2) we understand critical pointsof the C 1 (W 1,p (Ω)) functional J λ .following norm‖u‖ =Hereafter, we will denote by ‖ ‖ the( ∫ ∫ ) 1|∇u| p + a(x)|u| p pΩ∂Ωon W 1,p (Ω). As in the previous section, P Q and P ∗ stand for the followingfunctionals∫P (u) = ‖u‖ p , Q(u) =Now, we are able to state the followingΩ∫|u| q dx and P ∗ (u) =Ω|u| p∗ dx.Theorem 3.1 Let 1 < q < p, there exists ̂λ such that Problem (1.2) has atleast two positive solutions for λ ∈ (0, ̂λ).Proof. With slight changes in the proofs of the last section we obtain theresult.□

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