A nonlocal concave-convex problem with nonlocal mixed boundary data

The aim of this paper is to study a nonlocal problem with a mixed Dirichlet-Neumann exterior condition. We prove existence, nonexistence and multiplicity of positive energy solutions and describe the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data.

In this setting, Nsu can be seen as a Neumann condition of nonlocal type that is compatible with the probabilistic interpretation of the fractional Laplacian, as introduced in [20], and Bsu is a mixed Dirichlet-Neumann exterior datum. The main purpose of this work is to prove existence, nonexistence and multiplicity of positive energy solutions to problem (P λ ) for suitable ranges of λ and p and to understand the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data.
1. Introduction. In [20], the authors introduced a new nonlocal Neumann condition, which is compatible with the probabilistic interpretation of the nonlocal setting related to some Lévy process in R N . Motivated by this, we aim in this work to study a semilinear nonlocal elliptic problem with mixed Dirichlet-Neumann data. More precisely, we study existence and multiplicity of positive solutions to the following problem with 0 < q < 1 < p, N > 2s, λ > 0. In our setting, Ω ⊂ R N is a smooth bounded domain and (−∆) s is the fractional Laplacian operator, defined as See e.g. [23,24,17] and the references therein for more information about this operator. In this framework a N,s > 0 is a suitable normalization constant and the exterior condition can be seen as a nonlocal version of the classical Dirichlet-Neumann mixed boundary condition. As a matter of fact, here N s is the non-local normal derivative introduced in [20], given by N s u(x) = a N,s Ω u(x) − u(y) |x − y| N +2s dy, x ∈ R N \Ω.
Also, Σ 1 and Σ 2 are open sets in R N \Ω such that Σ 1 ∩Σ 2 = ∅ and Σ 1 ∪Σ 2 = R N \Ω. As customary, in (2) we denoted by χ A the characteristic function of a set A. We observe that, differently from the case of homogeneous Dirichlet conditions, the case of Neumann and mixed boundary conditions has not been much investigated in the fractional setting. This is due to the fact that the classical Neumann condition combines good geometrical properties (e.g. the normal derivative of the function vanishes, allowing symmetry and blow-up arguments) and analytic properties, while in the nonlocal case the consequences of (3) are much less intuitive and harder to deal with. This is indeed probably the first article devoted to the analysis of a nonlinear and nonlocal problem with mixed exterior data that involve the Neumann condition of [20]. We notice that recently a Hopf Lemma has been proved in [8] for such mixed exterior conditions.
Using an integration by parts formula stated in [20], one sees that problem (P λ ) can be set in a variational setting, since the requested solutions can be seen as critical points of the functional where |v| r dx and u + = max(u, 0).

A NONLOCAL CONCAVE-CONVEX PROBLEM 1105
Such problem, in the local case of the classical Laplacian, was extensively studied in the literature, especially after the seminal work of Ambrosetti, Brezis and Cerami [3]. Similar problems with a Dirichlet-Neumann datum were studied, for the subcritical case, in [15] and, in the critical case, in [22]. In the nonlocal framework, (that is, when s ∈ (0, 1)), with Dirichlet data, the problem was dealt with in [9] for the subcritical case and in [7,18,19] and [11] for the critical case. See also [28,29].
In [7] and [19], the authors use an extension method, introduced in [13], which allows them to reduce the problem to a local one. We stress that, in our case, because of the nonlocal Neumann part, we cannot use such extension and then we deal with the problem in an appropriate purely nonlocal, and somehow more general, framework. Moreover, to obtain our multiplicity result, we have to use an additional argument which was classically developed by Alama in [1].
For a series of motivations about nonlocal equations and fractional operators, see e.g. [12] and the references therein.
The paper is organized as follows: In Section 2, we introduce the functional setting to deal with problem (P λ ), as well as the notion of solution we will work with and some auxiliary results. Section 3 is devoted to prove the existence of minimal and extremal solutions. Finally in Section 4 we prove the existence of a second solution using Alama's argument.
2. Preliminaries and functional setting. We introduce in this section a natural functional framework for our problem and we give some related properties and some useful embedding results needed when we deal with problem (P λ ). According to the definition of the fractional Laplacian, see [17], [28], and the integration by parts formula, see [20] , it is natural to introduce the following spaces. We denote by H s (R N ) the classical fractional Sobolev space, endowed with the norm It is clear that H s (R N ) is a Hilbert space.
We recall now the classical Sobolev inequality that the proof can be found in [17]. See also [25] for an elementary proof. Proposition 1. Let s ∈ (0, 1) with N > 2s. There exists a positive constant S = S(N, s) such that, for any function u ∈ H s (R N ), we have where 2 * s = 2N N −2s . Definition 2.1. Let Ω be a bounded domain of R N . For 0 < s < 1, we define the space For u ∈ H s (Ω, Σ 1 ), we set The properties of this norm are described by the following result.
Proposition 2. The norm . in H s (Ω, Σ 1 ) is equivalent to the one induced by H s (R N ), and then (H s (Ω, Σ 1 ), , ) is a Hilbert space with scalar product given by Proof. For u ∈ H s (Ω, Σ 1 ), we set It is clear that H s (Ω, Σ 1 ) is a Hilbert space with the associated scalar product given by Notice that the completeness of H s (Ω, Σ 1 ) can be proved using exactly the same argument as in the proof of of Proposition 3.1 in [20]. Now, setting then the authors in [8] proved that λ 1 (Ω) > 0. As a consequence, the previous scalar product can be reduced to the following one Hence we can endowed H s (Ω, Σ 1 ) with the Gagliardo norm Now, the norm . in H s (Ω, Σ 1 ) is bounded by the one induced by H s (R N ), and so, using the Open Mapping Theorem, it holds that the norm . in H s (Ω, Σ 1 ) is in fact equivalent to the one induced by H s (R N ). Hence the result follows.
The following result justifies our choice of . .
The proof of this result is a direct application of the integration by parts formula, see Lemma 3.3 in [20].
In the rest of the paper, for the simplicity of typing, we shall denote the functional space introduced in Definition 2.1 by H s and we shall normalize the constant a N,s to be equal to 2. Now we give a Sobolev-type result for functions in H s .
Consider now the standard truncation functions given by and G k (u) = u − T k (u). In this setting, the following are some useful properties of H s -functions which are needed to get some regularity results for some elliptic problems in H s (see also Theorem 2.5 below).
Proof. The claim in (1) follows from the setting of the norm given in Definition 2.1. As for (2) and (3), we claim that, for any a, b ≥ 0 and any x ∈ R N , To check this, we can take which is the minimum value that T k (u) attains, and therefore (−∆) s T k (u)(x) ≤ 0 and N s T k (u)(x) ≤ 0. By combining these observations, we obtain (8). From (8) and Proposition 3 it follows that Using the normalization condition and by Propositions 3, we reach that In a similar way, Then, the claim in (2) follows from (10) and (9), while the claim in (3) follows from (11) and (9).
Let us now consider the following problem, where Ω is a bounded regular domain of R N , N > 2s, H −s is the dual space of H s and f ∈ H −s .
Definition 2.2. We say that u ∈ H s is an energy solution to (12) if where ( , ) represent the duality between H s and H −s .
Notice that the existence and uniqueness of energy solutions to problem (12) follow from the Lax-Milgram Theorem. Furthermore if f ≥ 0 then u ≥ 0. Indeed for u ∈ H s , thanks to Lemma 4, we know that u − = min{u, 0} ∈ H s . Taking u − as a test function in (13) it follows that u − = 0.
A supersolution (respectively, subsolution) is a function that verifies (13) with equality replaced by "≥" (respectively, "≤") for every non-negative test function in H s . Using a standard iterative argument we can easily prove the following result. Lemma 2.3. Assume that problem (12) has a subsolution w and a supersolution w, verifying w ≤ w. Then there exists a solution w satisfying w ≤ w ≤ w.
Here we prove some regularity results when f satisfies some minimal integrability conditions. To prove the boundedness of the solution we follow the idea of Stampacchia for second order elliptic equations with bounded coefficients. The interior Hölder regularity is a consequence of continuity properties, see [20], and the regularity results in [30]. Lemma 2.4. Let u be a solution to problem (12). If f ∈ L q (Ω), q > N 2s , then u ∈ L ∞ (Ω).
Proof. We follow here a related argument presented in [24]. See also [30] and [18] for related results. Let k > 0 and take ϕ = G k (u) as a test function in (13). Hence, thanks to Proposition 4, we get where A k = {x ∈ Ω : u > k}. Recalling (12), we obtain Applying Corollary 1 in the left hand side and Hölder inequality in the right hand side, we obtain we have that, and then, Since m > N 2s we have that Hence we apply Lemma 14 in [24] with ψ(σ) = |A σ | and the result follows.
Corollary 2. Let u be an energy solution of (12) and suppose that f ∈ L ∞ (Ω).
Proof. We claim that u is bounded in R N . Then one could apply interior regularity results for the solutions to (−∆) s u = 0 ∈ Ω and u = g in Ω c . See e.g. [30] and [26].
As a variation of Lemma 2.4, we point out that if f = f (x, u) and f has the following growth then, using a Moser iterative scheme, we can prove that: Theorem 2.5. Let u be an energy solution to problem (12) with f satisfies (16), then u ∈ L ∞ (Ω).
The following is a strong maximum principle for semi-linear equations, it will be used to separate minimal solution of problem (P λ ) for different values of the parameter λ, see [16].
Suppose furthermore that If there exists a point x 0 ∈ Ω at which v(x 0 ) = w(x 0 ), then v = w in the whole Ω.
Proof. Let φ = v − w and set By assumption x 0 ∈ Z φ . Moreover, thanks to the continuity of φ, we know that Z φ is closed. We claim now that Z φ is also open. Indeed, letx ∈ Z φ . Clearly φ ≥ 0 in R N , φ(x) = 0 and in view of (17). Accordingly, Hence φ vanishes identically in B (x) and then, for small, B (x) ⊆ Z φ . That is, we have proved that Z φ is open, and so, by the connectedness of Ω, we get that Z φ = Ω. Now we establish two important results for our purposes. The first result is a Picone-type inequality and the second is a Brezis-Kamin comparison principle for concave nonlinearities.
Theorem 2.6. Consider u, v ∈ H s , suppose that (−∆) s u ≥ 0 is a bounded Radon measure in Ω, u ≥ 0 and not identically zero, then, The proof of this result is based on a punctual inequality and follows in the same way as in [24]. As a consequence, we have the next comparison principle that extends to the fractional framework the classical one obtained by Brezis and Kamin, see [10].
Then u ≥ v in Ω.
The proof of this result is a slight modification of the proof of Theorem 20 in [24]. Finally, we will use the following compactness lemma to get strong convergence in the space H s .
Now, using Young's inequality, we obtain that As a consequence, v n → v strongly in H s .

3.
Proof of Theorem 1.1. In this section we prove Theorem 1.1. We split the proof into several auxiliary Lemmas. Let us begin by proving an existence result.
Proof. The main idea is to show that for λ small, the problem (P λ ) has a comparable bounded sub and supersolution. Let V be the unique positive solution to the problem Notice that the existence of V follows by using the Lax-Milgram theorem in the space H s , however the positivity of V follows form [8]. It is clear that V ∈ C α (Ω) for some α < 1. Let C = V ∞ , it is not difficult to show the existence of λ * > 0 such that for all λ < λ * , the inequality has a solution M > 0. Fix λ, M as above and define Thus v 1 is a supersolution to problem (P λ ).
We consider now the following problem Since q ∈ (0, 1), then setting it follows that M is achieved by a minimizer z. It is clear that z ≥ 0, then by Proposition 5 and Lemma 2.7, it follows that z > 0 and it is unique. In particular, z is the solution to problem (19). By Theorem 2.5, it holds that z ∈ L ∞ (Ω). Now setting z λ = λ 1 1−q z, then z λ is a solution to By the comparison result in Lemma 2.7, it holds that z λ ≤ v 1 . It is clear that z λ is a subsolution to problem (P λ ). Hence a monotonicity argument allows us to get the existence of a solution u λ to problem (P λ ) with z λ ≤ u λ ≤ v 1 . Then 0 < Λ < ∞.
We show now that Λ < ∞. Let λ be such that problem (P λ ) has a solutionū λ . By the comparison principle in Lemma 2.7, we get z λ ≤ū λ where z λ is the unique positive solution to problem (20). Let φ ∈ H s , then using Picone's inequality we obtain that This gives point (2) in Theorem 1.1.
We show now that for all 0 < λ < Λ, problem (P λ ) has a solution. This will be a consequence of the following lemma.
Then S is an interval.
Proof. Notice that S = ∅, thanks to Lemma 3.1. Let λ 1 ∈ S be fixed, we have just to prove that for all 0 < λ 2 < λ 1 , problem (P λ2 ) has a non trivial solution.
Since λ 1 ∈ S, then we get the existence of u 1 ∈ H s such that u 1 solves (P λ1 ). It is clear that u 1 is a supersolution to problem (P λ2 ). Recall that z is the unique solution to problem (19). Setting z 2 = λ in Ω, B s z 2 = 0 in R N \Ω.
By the comparison principle in Lemma 2.7, it holds that z 2 ≤ u 1 .
Since z 2 is a subsolution to problem (P λ2 ), then using a monotonicity argument we get the existence of u 2 ∈ H s such that z 2 ≤ u 2 ≤ u 1 and u 2 solves problem (P λ2 ). Thus λ 2 ∈ S and the result follows.
We now prove that (P λ ) possesses a minimal solution and we give some energy properties of such solutions. Lemma 3.4. For all 0 < λ < Λ, problem (P λ ) has a minimal solution u λ such that J λ (u λ ) < 0. Moreover the family u λ of minimal solutions is increasing with respect to λ.
Proof. Suppose that (P λ ) has a solution v λ for a given λ ∈ S. Define the sequence v n by v 0 = z λ , where z λ is the unique solution to problem (20). By the comparison result in Lemma 2.7, we have thatz ≤ ... ≤ v n−1 ≤ v n ≤ v λ and then, by Proposition 5, it follows that z λ < v n < v λ . So, using v n as a test function in (22), we get v n ≤ v λ . Hence there exists u λ ∈ H s such that v n u λ . Accordingly, since (−∆) s v n ≥ 0, using Lemma 2.8, we conclude that v n → u λ strongly in H s and u λ ≤ v λ . This shows that u λ is a minimal solution.
Then, by Lemma 2.7 and Proposition 5, we obtain the monotonicity of the family {u λ , λ ∈ (0, Λ)}. Henceforth, given λ ∈ (0, Λ), we use the notation u λ for the minimal solution. Let us define a(x) = λqu q−1 λ + pu p−1 λ and let µ 1 be the first eigenvalue of the following the problem Using closely the same argument as in the proof of Lemma 3.5 in [3], we can prove that It is clear that (24) is equivalent to Since u λ is a solution to (P λ ), testing the equation against u λ itself, we find that By (25), it follows that By inserting these relations into (4), we obtain that J λ (u λ ) < 0, as desired.
This gives point (1) in Theorem 1.1. Thus, to complete the proof of Theorem 1.1, we can now focus on the proof of point (3). To this end, we have the following result: Lemma 3.5. Problem (P λ ) has at least one solution if λ = Λ.
Proof. Let {λ n } be a sequence such that λ n Λ. We denote by u n ≡ u λn the minimal solution to problem (P λn ), then the sequence {u n } n is increasing in n. Since J λn (u n ) < 0, we get Then, it follows that {u n } is bounded in H s . Accordingly, we have that u n u * in H s , for some u * ∈ H s . Since {u n } n is increasing in n, using the fact that (−∆) s u n ≥ 0, recalling Lemma 2.8, we conclude that u n → u * strongly in H s . As a consequence, u * is a solution of (P λ ) for λ = Λ. Remark 1. If p ≤ 2 * s − 1 then using Theorem 2.5, we can easily prove that u * ∈ L ∞ (Ω), that means that u * is a regular extremal solution.
In view of Lemma 3.5, we obtain point (3) of Theorem 1.1. The proof of Theorem 1.1 is thus complete. 4. Proof of Theorem 1.2. In this section we prove the existence of a second positive solution to (P λ ).
Since p < N +2s N −2s , we observe that problem (P λ ) has a variational structure, indeed it is the Euler-Lagrange equation of the energy functional in (4). We note that J λ is well defined, it is differentiable on H s and for any ϕ ∈ H s , Thus critical points of the functional J λ are solutions to (P λ ).
To prove Theorem 1.2, we will use a mountain pass-type argument. The proof goes as follows. As in the local case, we can prove that the problem has a second positive solution for λ small. This follows using the mountain pass theorem. For this purpose it is essential to have a first solution which is a local minimum in H s . Let We define the functional J λ (u) = 1 2 u 2 − Ω F λ (u). Critical points of J λ correspond to solutions of (P λ ). Define the set A = {λ > 0 : J λ has a local minimum u 0,λ }.
It is clear that if λ ∈ A and w λ is a minimum of J λ in H s , then v = 0 is a local minimum of the functionalĴ where We can see thatĴ λ possesses the mountain pass geometry. Thus, let v 0 ∈ H s be such thatĴ λ (v 0 ) < 0 and define We have that c ≥ 0 and since p < 2 * s − 1, thenĴ λ satisfies the Palais-Smale condition. If c > 0, then using the Ambrosetti-Rabinowitz theorem we reach a non trivial critical point. If c = 0, then we use the Ghoussoub-Preiss Theorem, see [21].
As a consequence if we start with a local minimum of the functionalĴ λ , then we obtain a second critical point ofĴ λ , and hence a second solution to (P λ ).
Next, to show that problem (P λ ) has a second solution for all λ ∈ (0, Λ), we follow some arguments similar to those developed by Alama in [1] taking into consideration the nonlocal nature of the operator.
We prove first, using a variational formulation of the Perron's method, that the functional has a constrained minimum and then that this minimum is a local minimum in the whole H s . To this end, we use a truncation technique and some energy estimates.
It is clear that u 0 ∈ M and that M is a convex closed subset of H s . Since J λ0 is bounded from below in M and lower semi-continuous, then we get the existence of ϑ ∈ M such that J λ0 (ϑ) = inf u∈M J λ0 (u).
Let v be the unique solution to We have that J λ0 (v) < 0, and then ϑ = 0. As in Theorem 2.4 in [32], page 17, we conclude that ϑ is a solution to problem (P λ ). If ϑ = u 0 , then the proof of Theorem 1.2 is complete. Accordingly, we can assume that ϑ = u 0 . We show that ϑ is a local minimum of J λ0 .
For this, we argue by contradiction, and we assume that ϑ is not a local minimum of J λ0 . Then there exists a sequence {v n } ⊂ H s such that v n −ϑ H s → 0 as n → ∞ and J λ0 (v n ) < J λ0 (ϑ).
We define w n = (v n −ū) + and u n = max{0, min{v n ,ū}}. It is clear that u n ∈ M and ifū(x) ≤ v n (x).
Thus u n = v + n − w n . Let T n = {x ∈ Ω : u n (x) = v n (x)} and S n = supp w n ∩ Ω. Notice that supp v + n ∩ Ω = T n ∪ S n . We claim that |S n | → 0 as n → ∞.
To this end, let > 0,