ATTAINABILITY OF THE FRACTIONAL HARDY CONSTANT WITH NONLOCAL MIXED BOUNDARY CONDITIONS: APPLICATIONS

. The ﬁrst goal of this paper is to study necessary and suﬃcient conditions to obtain the attainability of the fractional Hardy inequality where Ω is a bounded domain of R d , 0 < s < 1, D ⊂ R d \ Ω a nonempty open set, N = ( R d \ Ω) \ D and s . The second aim of the paper is to study the mixed Dirichlet-Neumann boundary problem associated to the minimization problem and related properties; precisely, to study semilinear elliptic problem for the fractional Laplacian , that is, mixed boundary condition, Hardy inequality, doubly-critical problem. Ω , with N and D open sets in R d \ Ω such that N ∩ D = ∅ and N ∪ D = R d \ Ω, d > 2 s , λ > 0 and 1 < p ≤ 2 ∗ s − 1, 2 ∗ s = 2 d d − 2 s . We emphasize that the nonlinear term can be critical. The operators ( − ∆) s , fractional Laplacian, and N s , nonlocal Neumann condition, are deﬁned below in (7) and


1.
Introduction. The problems studied in this paper are motivated by some recent results that we summarize below.
Theorem. (Fractional Hardy inequality). Assume that s ∈ (0, 1) is such that 2s < d, then for all u ∈ C ∞ 0 (R d ), the following inequality holds, a d, whereû is the Fourier transform of u and . (

2)
Moreover the constant Λ is optimal and is not attained.
Note that the constant a d,s is defined by the first identity in (1) and is given by (see for instance, [12,15,21,27]). The optimal constant defined in (2) coincides for every bounded domain Ω containing the pole of the Hardy potential. More precisely, if 0 ∈ Ω, then for all u ∈ C ∞ 0 (Ω) we have a d,s 2 DΩ |u(x) − u(y)| 2 |x − y| d+2s dx dy ≥ Λ Ω u 2 |x| 2s dx, where The optimality of Λ here follows by a scaling argument. The other starting points for the problems considered in this work are some results obtained in the articles [17], [25] and [16].
In [17] the authors consider a natural Neumann condition in the sense that Gauss and Green integration by parts formulas hold for such condition. More precisely, if Ω is a bounded open set in R d with suitable regularity, then the Neumann problem for the fractional Laplacian takes the form, where (−∆) s is the fractional Laplacian operator defined by (−∆) s u(x) = a d,s P.V.
a d,s > 0 being the normalization constant defined in (3) and See e.g. [24], [26], [15] and the references therein for more properties of this operator. Notice that as a consequence of the analysis of the sequence of eigenvalues with Neumann condition done in [17], we reach that the best constant for the Hardy inequality with Neumann condition is 0 and it is attained by any constant function.
With this meaning for the Neumann condition, the authors in [25] studied the behavior of the eigenvalues for mixed Dirichlet-Neumann problems in terms of the boundary conditions. In particular, they proved a necessary and sufficient condition for the convergence of the first eigenvalue of mixed problems to 0, the principal eigenvalue for Neumann problem.
These previous results are the inspiration for our main goal in this paper: to study the attainability of the fractional Hardy constant with mixed boundary condition.
More precisely, let Ω ⊂ R d be a regular bounded domain containing the origin and consider N and D to be two open sets of R d \Ω such that N ∩ D = ∅ and N ∪ D = R d \Ω.
A such pair (D, N ) will be called a Dirichlet-Neumann configuration, D-N configuration to be short.
Notice that in the whole paper and for simplicity of typing, we set dν = dx dy |x − y| d+2s .
We define where E s (Ω, D) = u ∈ H s (R d ) : u = 0 in D . The above minimizing problem is strongly related to the next eigenvalue problem in Ω, The mixed boundary condition B s for the D-N configuration given by D and N open sets in R d \Ω such that N ∩ D = ∅ and N ∪ D = R d \ Ω, is defined by and N s is defined in (8). As customary, in (12), we denoted by χ A the characteristic function of a set A.
In the local case s = 1, we can mention the works [3] and [4] where the authors have found some conditions of monotonicity that ensure the attainability or not of the Hardy constant. As a consequence they analyze a doubly-critical problem related to a mixed Sobolev constant. In [4] the authors deal with the same type of problem associated to elliptic operators in divergence form associated to the Caffarelli-Kohn-Nirenberg inequalities.
We will extend the previous results to the fractional Laplacian framework without any condition of monotonicity and then we get a stronger results than in the local case.
It is worthy to point-out that a kind of nonlocal mixed boundary conditions for the spectral fractional Laplacian was defined recently by several authors, see for example [13], where the authors used the Caffarelli-Silvestre extension to define a suitable nonlocal Neumann boundary condition. We refer also to [22] for other type of Neumann condition.
The main result in this work related to the attainability of the constant Λ N defined in (10), is the following characterization, that is deeply related to the results in [25].
Notice that in the local case one of the main tools to analyze the compactness of the minimizing sequence is to use a suitable Concentration-Compactness argument that allows to avoid any concentration in Ω or at the boundary of Ω. In the nonlocal setting we will consider an alternative approach. Other point that gives the difference between the local and the nonlocal case is the fact that in the nonlocal case, the set N can be unbounded and little is known about the regularity of the solution in this set.
The second goal in this paper is the analysis of some semilinear problems, even with critical growth. There is a large literature about semilinear perturbations of the fractional Laplacian with Dirichlet boundary conditions. Notice that the subcritical concave-convex problem with mixed boundary conditions is studied in [2]. However the doubly critical problem has been only considered in the case of the whole Euclidean space, see [16] for details.
Consider the Sobolev constant, S N , defined in Proposition 4, and the critical constant studied in [16], where it is analyzed the problem in the whole space R d . The relevant result with respect to the solvability of the doubly critical problem with mixed boundary condition is the following Theorem which gives the condition of existence in terms of S N and S λ . More precisely we state the result. N ) a D-N configuration and assume that λ ∈ (0, Λ N ). Define Then if T λ,N < min{S λ , S N }, the problem has a nontrivial solution.
See also Theorem 4.4 in the last section of the paper. The paper is organized as follows. In Section 2, we introduce some analytical tools needed to study the problem (P λ ), such as the natural fractional Sobolev space associated to problem (P λ ), some classical functional inequalities, and the adaptation of a Picone inequality type obtained in [26].
The Hardy constant for mixed problems is treated in Section 3, where we prove Theorem 1.1. The proof is more involved than in the local case and we prove some previous sharp estimates that we need to obtain the main result. In Subsection 3.1 we give sufficient condition to guarantee the attainability of Λ N ; the non-attainability is analyzed in Subsection 3.2. In both cases we give explicit examples where these the attainability and the non-attainability are realized.
In the last section, Section 4, among others results, we prove Theorem 1.2, that is, we study the solvability of the doubly-critical problem.
2. Preliminaries and functional setting. We introduce in this section the natural functional framework for our problem and we give some properties and some embedding results needed when we deal with problem (P λ ).
According to the definition of the fractional Laplacian, see [15], and the integration by parts formula, see [17], it is natural to introduce the following spaces. We denote by H s (R d ) the classical fractional Sobolev space, endowed with the norm where dν is defined in (9). It is clear that H s (R d ) is a Hilbert space.
We recall now the classical Sobolev inequality that is proved for instance in [15]. See also [28] for an elegant geometrical proof. Proposition 1. Let s ∈ (0, 1) with d > 2s. There exists a positive constant S = S(d, s) such that, for any function u ∈ H s (R d ), we have where 2 * s = 2d d−2s . Beside to the Hardy inequalities (1) and (4), in the case of bounded domain Ω, we have the next regional version of the Hardy inequality whose proof can be found in [1].

Proposition 2.
Let Ω be a bounded regular domain such that 0 ∈ Ω, then there exists a constant C ≡ C(Ω, s, d) > 0 such that for all u ∈ C ∞ 0 (Ω), we have Since we are considering a problem with mixed boundary condition we need to specify the space where the solutions belong. For u ∈ E s (Ω, D), we set where D Ω is defined in (5) and the measure dν is defined in (9). The properties of this norm are described by the following result. We refer to [17], [8] and [25] for the proof and other properties of this space.
moreover there exists a positive constant C(Ω) such that the next Poincaré inequality holds: As a consequence of the definition of E s (Ω, D) and using the extension result proved in [17], we get the next Sobolev inequality in the space E s (Ω, D).
The proof of this result follows by the application of the integration by parts formula given in Lemma 3.3 of [17].
We give the definition of weak solution for the elliptic problem with mixed boundary condition.
where f ∈ (E s (Ω, D)) , the dual space of E s (Ω, , D). We say that u is a weak solution to problem (20) if It is clear that the existence of u follows using the classical Lax-Milgram Theorem, see [25] for more details.
In order to obtain some a priori estimates, we will use the next Picone type inequality that is an extension of the corresponding inequality in H s 0 (Ω) obtained in [26]. For the reader convenience we give the proof.
In particular, if we have equality in (21), then there exists a constant C such that v = Cu in R d .
Proof. Notice that for u, v as in the hypotheses of the Theorem we have the following simple identity Integrating the previous identity with respect to dν, defined in (9), we conclude. Finally, if we have the equality in (21), then we conclude that the result follows.
3. Analysis of the mixed Hardy optimal constant. Consider Ω ⊂ R d a bounded domain and D, N ⊂ R d \ Ω a D-N configuration. In this section we will analyze the condition for the attainability of the mixed Hardy constant defined by We start by proving the following result.
Proof. We begin by proving the positivity of Λ N . Let u ∈ E s (Ω, D) and fix δ > 0 In what follows we denote by C or C(Ω) any positive constant that depends on Ω, d, s, that is independent of u and that can change from line to line.
It is clear that Since 1 − ϕ = 0 in B δ (0), then using the Poincaré inequality in (17), we conclude that We deal now with the term Ω (uϕ) 2 |x| 2s dx. Since uϕ ∈ H s 0 (Ω), then by the Hardy inequality (4), we obtain that The immediate algebraic identity In first place, it is clear that Respect to J 2 , since Ω is a bounded domain, it holds that Since |ξ|≤ε 1 |ξ| d+2s−2 dξ < ∞, using the Poincaré inequality in Proposition 3, we get By using Young inequality, we reach that Therefore, combining the above inequalities, we conclude that Going back to (23), and by using the estimates (24), (25), we conclude that Hence Λ N > 0 and then the first affirmation follows.
The main result in this section is Theorem 1.1. We split the proof into two parts contained in the following two subsections.
Next we proceed to estimate I 2 2 (n). We have Finally, we consider the term I 3 (n).
Notice that Using now Young inequality, we get that Choosing ε small enough, we obtain that Therefore, from (27), it holds that Going back to (26), we conclude that which is a contradiction with the hypothesis Λ N < Λ. Henceū = 0 and then the claim follows.
To show that Λ N is achieved we will use the Ekeland variational principle, see [18]. Then up to a subsequence, it holds that Let ϕ ∈ E s (Ω, D), by duality argument we obtain that Thusū solves the problem Choosingū as a test function in (30), we obtain 3.1.1. Properties of the spectral value Λ N if Λ N < Λ. In this subsection we treat the case Λ N < Λ, thus Λ N is achieved and we prove that it behaves like a principal eigenvalue of the mixed elliptic problem with the Hardy weight.
To start, we begin by giving some configurations of (D, N ) for which the constant Λ N is reached. By the previous results it suffices to prove that Λ N < Λ. This last inequality is a straightforward consequence of some results contained in [25]. For the reader convenience we explain below some details.
We say that Ω is an admissible domain if it is a C 1,1 and it satisfies the exterior sphere condition. Now, let consider sequences of sets Following closely the same argument as in [25], we obtain the next result.
In the next result we show the relevant spectral properties of Λ N when it is reached.
Theorem 3.4. Assume that Λ N < Λ, then Λ N is an isolated and simple eigenvalue, that is: 1. If v is an other solution to problem (31) with λ = Λ N , then v = Cū, C ∈ R.
Proof. Let us begin by proving the first point. Suppose that v is another solution to problem (31) with λ = Λ N . If v ≥ 0, then using the Picone alternative in Theorem 2.3, we conclude that v = Cū for some C ≥ 0. Now, suppose that v change sign. Since v is a weak solution in the sense of Definition 2.2 to the problem (31), in particular, v ∈ E s (Ω, D) and then v ± ∈ E s (Ω, D). Thus, by using v + (respectively v − ) as a test function in the equation of v, we obtain that Hence v ± realizes the minimum in (22) and then there are nonnegative solutions to problem (31) with λ = Λ N . Hence there exists C ± ≥ 0 such that v ± = C ±ū , thus We next prove (2), that is, the eigenvalue Λ N is isolated. Assume the existence of a sequence {(λ n , u n )} n ⊂ (Λ N , ∞) × E s (Ω, D) such that λ n ↓ Λ N and u n solves the problem (31) with λ = λ n . Without loss of generality we can assume that Ω u 2 n |x| 2s dx = 1.

3.2.
If Λ N = Λ, then Λ N is not attained. We prove the following result that complete the proof of Theorem 1.1.

Proposition 7.
In the hypotheses of Theorem 1.1 assume that Λ N = Λ, then Λ N is not attained.
Proof. We argue by contradiction. Assume that Λ N is attained in E s (Ω, D), then there exists u ∈ E s (Ω, D) such that in Ω, Let B r (0) ⊂⊂ Ω and define v to be the unique solution of the problem in Ω, From Lemma 3.9 in [5], we obtain that we know that w( which is a contradiction with the fact that w ∈ H s (Ω). Hence the result follows.

3.2.1.
Examples for which we find Λ N = Λ. In this subsection we give some geometrical condition to ensure that Λ N = Λ. We have the next result.
and suppose that N s w(x) ≥ 0 for all x ∈ N , then Λ N = Λ. Moreover, the problem in Ω, has no solution.
Proof. We argue by contradiction. Assume that N s w(x) ≥ 0 for all x ∈ N and that Λ N < Λ. Then by Theorem 1.1, we get the existence of u 1 ∈ E s (Ω, D) a positive solution to problem (36).
According to the symmetry of last two terms of the above identity, we obtain that a d,s 2 Notice that a d,s 2 Therefore, Hence, it holds that Thus, if N s (w(x)) ≥ 0 for all x ∈ N , it follows that v 1 = 0 and we get a contradiction. Hence we conclude.
Here we give an explicit bounded domain where the above situation holds. Define the set Ω = Ω 1 ∪ Ω 2 ∪ Ω 3 where  To prove that Λ N = Λ, we will show the existence of ε 0 such that if ε ≤ ε 0 , then N s w(x) ≥ 0 for all x ∈ N . Recall that w(x) = |x| − d−2s 2 , for simplicity of typing we set θ 0 = d−2s 2 , then Let us begin by estimating J 1 . Setting y = |y|y and x = |x|x , it holds that |x| , then following closely the radial computation as in [19], it follows that Choosing ε << η, there results that ε |x| ≤ ε η << 1, hence We deal now with J 2 . Without loss of generality we will assume that ε < min{ η 4 , m 4 } and fix = min{ It is clear that for all x ∈ N and for all y ∈ Ω 2 , we have |x − y| ≥ . Thus Since |y| −θ0 ∈ L 1 loc (R d ), then by the Dominated Convergence Theorem it holds We deal now with J 3 . Following closely the computation of J 1 , we reach that Choosing β >> 2 and combining the above estimates, we conclude that Hence we conclude.
We have also the example described by the Figure 2, where the constant Λ N = Λ and then it is not attained. We leave the details to the reader. 4. Semilinear mixed problem involving the Hardy potential. In this section we assume that Λ N < Λ, that is, Λ N is the principal eigenvalue for the corresponding mixed problem. We will consider the following nonlinear problem where 1 < p ≤ 2 * s − 1 and λ < Λ N . 4.1. Subcritical problems, 1 < p < 2 * s −1. The next result is a direct consequence of Theorem 3.4 and the classical Rabinowitz bifurcation Theorem, see [29].
More interesting is the following problem. Assume now that λ ∈ (Λ N , Λ) and define where p ∈ (1, 2 * s − 1). It is clear that I λ,p < 0, however we have the next result. Theorem 4.2. Assume that λ ∈ (Λ N , Λ) and 1 < p < 2 * s − 1, then I λ,p < 0, is finite and it is achieved. As a consequence the problem in Ω, has a positive solution.
Proof. We divide the proof into two steps.
Since p + 1 > 2, then On the other hand we have We estimate the last integral. By a direct computation it holds that By the elementary identity, and using Young inequality, we obtain that Ω Ω u 2 (y)(ψ(x) − ψ(y)) 2 dν.
Step 2. I λ,p is attained. Define , it is clear that I λ,n ↓ I λ,p as n → ∞. Hence I λ,n < 0 for n ≥ n 0 .
Since p + 1 < 2 * s , then using a variational argument we get that I λ,n is achieved. Hence there exists u n ∈ E s (Ω, D) that satisfies in Ω, with u n L p+1 (Ω) = 1 We claim that {u n } n is bounded in the space E s (Ω, D). Since then by (45), it follows that Thus DΩ (u n (x) − u n (y)) 2 dν ≤ C for all n and the claim follows.
Therefore, there exists u 0 ∈ E s (Ω, D) such that u n u 0 weakly in E s (Ω, D) and strongly in L p+1 (Ω). Hence u 0 L p+1 (Ω) = 1 and then u 0 ≡ 0. Notice that by the weak convergence we obtain that u 0 is a weak solution to (39).
We claim now that u 2 n |x| 2s → u 2 0 |x| 2s strongly in L 1 (Ω). Define w n = u n − u 0 , it is clear that w n 0 weakly in E s (Ω, D) and w n → 0 strongly in L p+1 (Ω). As in the previous step, we have DΩ (w n (x) − w n (y)) 2 dν ≥ DΩ (ψw n )(x) − (ψw n )(y) Hence Ω w 2 n |x| 2s dx = o(1) and the claim follows. Combing the above estimates we reach that u n → u 0 strongly in E s (Ω, D) and thus u 0 realize I λ,p . Hence up to a positive constant, cu 0 solves problem (39), then we conclude.

4.2.
Doubly-critical problem. In this subsection we discuss conditions for the existence and the non existence to the following double critical problem where λ ∈ (0, Λ N ) according to the D-N configuration. If Ω = R d , problem (46) is related to the next constant (47) The problem in the whole Euclidian space R d has been studied in [16]. From the result of [16] we know that the constant S λ is independent of Ω containing the pole of the Hardy potential.
In the same way we consider for a D-N configuration the constant T λ,N defined by (48) It is clear that if T λ,N is achieved, then problem (46) has a nontrivial solution.