THE NEHARI MANIFOLD FOR A FRACTIONAL LAPLACIAN EQUATION INVOLVING CRITICAL NONLINEARITIES

. We study the combined eﬀect of concave and convex nonlinearities on the numbers of positive solutions for a fractional equation involving critical Sobolev exponents. In this paper, we concerned with the following fractional equation where 0 < s < 1, λ > 0, 1 ≤ q < 2, 2 ∗ s = 2 N N − 2 s , 0 ∈ Ω ⊂ R N ( N > 4 s ) is a bounded domain with smooth boundary ∂ Ω, and f, g are nonnegative contin- uous functions on ¯Ω. Here ( − ∆) s denotes the fractional Laplace operator.


(Communicated by Changfeng Gui)
Abstract. We study the combined effect of concave and convex nonlinearities on the numbers of positive solutions for a fractional equation involving critical Sobolev exponents. In this paper, we concerned with the following fractional equation where 0 < s < 1, λ > 0, 1 ≤ q < 2, 2 * s = 2N N −2s , 0 ∈ Ω ⊂ R N (N > 4s) is a bounded domain with smooth boundary ∂Ω, and f, g are nonnegative continuous functions onΩ. Here (−∆) s denotes the fractional Laplace operator.
In recent years, considerable attention has been given to nonlocal diffusion problems, in particular to the ones driven by the fractional Laplace operator. One of the reasons for this comes from the fact that this operator naturally arises in several physical phenomena like flames propagation and chemical reactions of liquids, and in population dynamics and geophysical fluid dynamics. It also provides a simple model to describe certain jump lévy processes in probability theory. For more details and applications, see [3,4,5] and the references therein.
For s = 1, equation (2) become the following semi-linear elliptic problem involving concave-convex nonlinearities −∆u + µu = λf (x)|u| q−2 u + |u| p−2 u, x ∈ Ω, where µ, λ ≥ 0, 1 ≤ q < 2, and f is a continuous function onΩ. Ambrosetti, Brezis and Cerami in [2]( µ = 0, f ≡ 1, 2 < p ≤ 2 * = 2N N −2 ) has at least two positive solutions for λ ∈ (0, Λ), and even more it has no positive solutions if λ > Λ. One can also see that the result from the bibliography of paper in [4,3,5]. Wu [29](µ = 0, f ∈ C(Ω) and changes sign, 2 < p < 2 * ) showed that equation (3) has at least two positive solutions for λ sufficiently small. Lin in [20] show that for λ ∈ (0, Λ), there exists a least one positive solution, and for different value of q, there is k + 1 positive solutions(k is integer). Any other related results about the subcritical case readers can refer [9,18,19,31] and the references therein. Especially, using the idea of category and Bahri-Li's minimax argument, Adachi and Tanaka [1] consider the following equation where g(x) ≡ 1, g(x) ≥ 1 − Ce −(2+δ)|x| for some C and δ, and sufficiently small f H −1 > 0, they admit at least four positive solutions in R N . For the semilinear fractional elliptic equation This type of operators naturally arises in physical situations such as thin obstacle problem, optimization, water waves and so on, see [16]. We refer to [13,22,23,21,25,28,30] for the subcritical case and to [4,3,24,27] for the critical case. In paper [8], Caffarelli and Silvestre gave a new formulation of the fractional Laplacian through Dirichlet-Neumann maps. This is extensively used in the recent literature since it allows to transform nonlocal problems to local ones, which permits using variational methods. For example, Barrios, Colorado, de Pablo and Sanchez [4] used the idea of the s-harmonic extension and studied the effect of lower order perturbations in the existence of positive solutions of (4). In [7], Cabre and Tan defined (−∆) 1 2 through the spectral decomposition of the Laplacian operator in Ω with zero Dirichlet boundary conditions, with classical local techniques, they established existence of positive solutions for problems with subcritical nonlinearities, regularity and L ∞ -estimates for weak solutions.
Very recently, Colorado, de Pablo, and Sanchez [15] studied the following nonhomogeneous fractional equation involving critical Sobolev exponent and proved the existence and multiplicity of solutions under right conditions on the size of f (q = 0). For q = 0, Barrios e.t. in [3] show under the different boundary which u = 0 in R N \ Ω, for 0 < q < 1, the equation admits no solution, at least one solution and at least two solution in different condition of λ. On the other hand, for 1 < q < 2 * s − 1, equation (5) admits at least one solution. For s = 1 2 , Carboni and Mugnai consider the subcritical case, they show λ < λ * , equation (5) does not admit non-trivial solution, and λ >λ, equation (5) has at least two non-trivial solutions, one is negative, one is positive. Also some relevant results about systems in [12,17] and the references therein.
The purpose of this paper is to study equation (2) in the critical case. Using variational methods and Nehari manifold decomposition, we prove the equation admits at least a positive solution when λ belongs to a certain subset of R. We adopt the spectral definition of the fractional Laplacian in a bounded domain based upon a Caffarelli-Silvestre type extension, and not the integral definition. Here we study a Dirichlet problem with critical growth, characterized by the following features: the equation contains a positive perturbation term with subcritical growth; moreover, a positive nonconstant coefficient g(x) multiplies the term with critical exponent.
(h 2 ) There exist k points a 1 , a 2 , · · · , a k in Ω such that Then we have the following theorems. This paper is organized as follows. In Section 2, we introduce the variational setting of problem (2) and present some preliminary results. In Section 3, we give some properties about the Nehari manifold and prove Theorem 1.1. In Section 4, we construct the k-compact Palais-Smale sequences which are suitably localized in correspondence of the k maximum points of g and prove Theorem 1.2.
2. Some preliminary facts. In this section, we collect some preliminary facts in order to establish the functional setting. First of all, let us introduce the standard notations for future use in this paper. We denote the upper half-space in R N +1 by Let Ω ⊂ R N be a smooth bounded domain. Denote the cylinder with base Ω and its lateral boundary by ∂ L C Ω := ∂Ω × (0, ∞). The powers (−∆) s with zero Dirichlet boundary conditions are defined via its spectral decomposition, namely where (ρ j , ϕ j ) is the sequence of eigenvalues and eigenfunctions of operator (−∆) s in Ω under zero Dirichlet boundary data and b j are the coefficients of u for the base {ϕ j } ∞ j=1 in L 2 (Ω). In fact, the fractional Laplacian (−∆) s is well defined in the space of functions and u H s 0 (Ω) = (−∆) s 2 u L 2 (Ω) . The dual space H −s (Ω) is defined in the standard way, as well as the inverse operator (−∆) −s . Definition 2.1. We say that u ∈ H s 0 (Ω) is a solution of (2) if the identity holds for all ϕ ∈ H s 0 (Ω). Associated with problem (2), we consider the energy functional The functional is well defined in H s 0 (Ω), moreover, the critical points of the functionalJ λ correspond to solutions of (2). We now conclude the main ingredients of a recently developed technique used in order to deal with fractional powers of the Laplacian. To treat the nonlocal problem (2), we shall study a corresponding extension problem, which allows us to investigate problem (2) by studying a local problem via classical variational methods. We first define the extension operator and fractional Laplacian for functions in H s 0 (Ω). We refer the reader to [4,3,6, 10] and the references therein.
Definition 2.2. For a function u ∈ H s 0 (Ω), we denote the s-harmonic extension w = E s (u) to the cylinder C Ω as the solution of the problem and The extension function w(x, y) belongs to the space  The extension operator is an isometry between H s 0 (Ω) and X s 0 (Ω), namely u H s 0 (Ω) = E s (u) X s 0 (CΩ) , for any u ∈ H s 0 (Ω).

CONCAVE-CONVEX NONLINEARITIES 2265
With this extension we can reformulate (2) as the following local problem where ∂w ∂ν s := −k s lim y→0 + y 1−2s ∂w ∂y , and w ∈ X s 0 (C Ω ) is the s-harmonic extension of u ∈ H s 0 (Ω). An energy solution to this problem is a function w ∈ E s 0 (C Ω ) satisfying (8), then u = w(·, 0), defined in the sense of trace, belongs to the space H s 0 (Ω) and it is a solution of the original problem (2). The associated energy functional to the problem (8) is denoted by Critical points of J λ in X s 0 (C Ω ) correspond to critical points ofJ λ (u) : H s 0 (Ω) → R N . In the following lemma we list some relevant inequalities from [6].
for some positive constant C = C(r, s, N, Ω). Furthermore, the space X s 0 (C Ω ) is compactly embedded into L r (Ω), for every r < 2 * s . Remark 1. When r = 2 * s , the best constant in (9) is denoted by S, that is It is not achieved in any bounded domain and for all z ∈ X s (R N +1 + ), S is achieved for Ω = R N by function w ε which are the s-harmonic extension of and let W be the extension of U ( [4,6]). Then is the extreme function for the fractional Sobolev inequality (11). The constant S given in (10) takes the exact value and it is achieved for Ω = R N by the functions w ε . Without loss of generality, we may assume k s = 1.
3. The Nehari manifold. It is useful to consider the functional on the Nehari manifold It is clear that all critical points of J λ must lie in M λ , and as we will see below, local minimizers on M λ are actually critical points of J λ . We have the following results.
Lemma 3.1. The energy functional J λ is coercive and bounded below on M λ .
Proof. For w ∈ M λ , by Hölder inequality and the Sobolev embedding theorem, we get Hence, we have that J λ (w) is coercive and bounded below on M λ .
Using (h 1 ), the Hölder and the Sobolev inequalities, we get By direct computation, we obtain Combining (16) and (17), we have which is a contradiction.
Proof. The proof is similar to Wu [29].
4. Existence of k+1 solution. In this section, we shall detect the range of value β for which (P S) β condition holds for the functional J λ . First of all, we want to show that J λ satisfies the (P S) β condition in X s 0 (C Ω ) for β ∈ (−∞, s N S N 2s −C 0 λ 2 2−q ). We shall have the following preliminary results.
Then J λ (w) = 0 in (X s 0 (C Ω )) −1 and there exists a positive constant Consequently, we get Then, combibing (21), the Young inequality and the Sobolev embedding theorem, we obtain, Then |β| + c n + d n w n X s Hence there exists w ∈ X s 0 (C Ω ) such that w n (x, y) w(x, y), weakly in X s 0 (C Ω ), w n (x, 0) → w(x, 0), a.e in Ω, Since J λ (w n ) = β + o n (1) and J λ (w n ) = o n (1), by J λ (w) = 0, we deduce that g(x)|w n − w| 2 * s dx, Now, we assume that Applying the Sobolev inequality, we obtain Then l ≥ Sl which is a contradiction. Hence, l = 0, that is w n → w strongly in X s 0 (C Ω ).
Recall that the best Sobolev constant S is defined as , then the extension of U ε,a i has the form Without loss of generality, we may assume that 0 ∈ Ω. We define the cut-off function φ ∈ C ∞ 0 (C Ω ), 0 ≤ φ ≤ 1 and for small fixed ρ > 0 where B ρ = (x, y) : |x − a i | 2 + y 2 < ρ 2 , y > 0 , we take ρ so small that B 2ρ ⊂ C Ω .
Define w i ε = φW i ε ∈ X s 0 (C Ω ) for ε > 0 small enough. From now on, we assume that N N −2s < q < 2 and N > 4s. Proof. By an argument similar to the proof of [4], we get We notice that Then, one has that

This implies that
Taking ε so small that Cε N U For N N −2s < q < 2, N > 4s and ε < ρ0 Next, we consider the functional J 0 : X s 0 (C Ω ) → R N defined by Step I. Show that sup By (h 2 ), we get that as Using (23) and (24), we obtain that by (25), we get that Step II. Choose a positive number Λ 1 < qΛ 2 such that 2s N S N 2s − C 0 λ 2 2−q > 0, for any λ ∈ (0, Λ 1 ).
By (h 1 ), we get for all t ≥ 0.
We know that where E s (u) = w and χ : By Lemma 2.5, we have that t i ε w i ε ∈ M − λ , then we have the following.
Next, in order to show that β i λ are (P S)−value. Associated with the critical problem, we define the energy functional I : X s 0 (C Ω ) → R, , w = 0 . Now we state the following global compactness result for I, which can be proved by the same argument in [26]. That is now standard, so we do not give a proof here.
It is well known that the best Sobolev constant S is independent of the domain and is never achieved except when Ω = R N . Moreover γ(Ω) = γ(R N ) = s N S N 2s .
Next, there is a sequence {s n } ⊂ R + such that s n w n ∈ M(Ω), then s n = 1 + o n (1) and I(s n w n ) = γ(Ω) + o n (1). Using the same argument as in [14], every minimizing sequence in M(Ω) of γ(Ω) is a (P S) γ(Ω) sequence in X s 0 (C Ω ) for I. Since inf v∈MΩ I(v) = γ(Ω) is not achieved, applying the Palais-Smale decomposition Lemma 4.5, we get that there exist sequences ε n > 0 and {x n } ⊂ Ω such that 1 εn dist(x n , ∂Ω) → ∞ as n → ∞ as n → ∞ and where U is the extremal function for S introduced in (10), is a solution of the following equation Since Ω is a bounded domain and {z n } ⊂ Ω, then ε n → 0. Suppose the subsequence z n → z 0 ∈Ω as n → ∞, we claim that z 0 ∈ K. By the Lebesgue dominated convergence theorem, then U ( z − x n ε n )) 2 * s dz + o n (1) = g(z 0 )S N 2s , that is z 0 ∈ K.