Nonlocal critical growth elliptic problems with jumping nonlinearities

In this paper we study a nonlocal critical growth elliptic problem driven by the fractional Laplacian in presence of jumping nonlinearities. In the main results of the paper we prove the existence of a nontrivial solution for the problem under consideration, using variational and topological methods and applying a new linking theorems recently got by Perera and Sportelli in [10]. The existence results provided in this paper can be seen as the nonlocal counterpart of the ones obtained in [10] in the context of the Laplacian equations. In the nonlocal framework the arguments used in the classical setting have to be refined. Indeed the presence of the fractional Laplacian operator gives rise to some additional difficulties, that we are able to overcome proving new regularity results for weak solutions of nonlocal problems, which are of independent interest.


Introduction
Fractional and nonlocal operators appear naturally in many different fields, see, for instance, [2,3,5,21] and the references therein.This is one of reason why nonlocal fractional problems are widely studied in the current literature.
From a mathematical point of view, nonlocal fractional Laplacian problems (and their generalizations to integrodifferential operators) have been considered in many different contexts and different results got in the classical context of uniformly elliptic equations have been extended to this nonlocal framework, see for instance the monograph [9] for superlinear and subcritical cases, critical ones, and many others.
Motivated by the recent paper [10], where the authors studied a critical growth elliptic problem with jumping nonlinearities, in this paper we consider its nonlocal counterpart.Precisely, here we study the existence of nontrivial solutions to the following nonlocal critical growth elliptic problem with a jumping nonlinearity where Ω is a open bounded subset of R N with Lipschitz boundary satisfying the exterior ball condition, (−∆) s is the fractional Laplacian operator defined on smooth functions by (−∆) s u(x) = 2 lim (1. 2) It was shown in Servadei [13,14] and Servadei and Valdinoci [17,19] that this problem has a nontrivial solution in each of the following cases: (i) N > 4s and λ > 0, (ii) N = 4s and λ > 0 is not a Dirichlet eigenvalue of (−∆) s in Ω, (iii) 2s < N < 4s and λ > 0 is sufficiently large.
This extends to the fractional setting the well-known results of Brézis and Nirenberg [4] and Capozzi et al. [6] for critical Laplacian problems (see also Ambrosetti and Struwe [1] and Costa and Silva [7]).
The set Σ((−∆) s ) consisting of points (a, b) ∈ R 2 for which the problem has a nontrivial solution is called the Dancer-Fučík spectrum of (−∆) s in Ω. Denoting by (λ l ) the sequence of Dirichlet eigenvalues of (−∆) s , the spectrum Σ((−∆) s ) contains the points (λ l , λ l ) since problem (1.3) reduces to the Dirichlet eigenvalue problem for (−∆) s when a = b.Moreover, it follows from the abstract results in Perera and Schechter [11,Chapter 4] that in the square Σ((−∆) s ) contains two strictly decreasing curves such that the points in Q l that are either below the lower curve C l or above the upper curve C l are not in Σ((−∆) s ), while the points between them may or may not belong to Σ((−∆) s ) when they do not coincide (see Theorem 2.2).
The main results of the paper are the following.
for some l ≥ 2, then problem (1.1) has a nontrivial solution.
for some l ≥ 2, then problem (1.1) has a nontrivial solution.
The classical linking arguments used to obtain nontrivial solutions of problem (1.2) in [13,14,17] rely on the decomposition of H s 0 (Ω) into eigenspaces of (−∆) s .These arguments are not suitable for proving Theorems 1.1 and 1.2 since problem (1.3) is nonlinear and therefore its solution set is not a linear subspace of H s 0 (Ω) when (a, b) ∈ Σ((−∆) s ).
We will prove our existence results using an abstract existence result for problems with jumping nonlinearities and new linking theorems based on a nonlinear splitting of the underlying space that were recently proved by Perera and Sportelli [10].We will recall these results in Section 2 (see Theorems 2.3 and 2.4).
The proof of Theorems 1.1 and 1.2 are in the line of the results got in [10], even if additional difficulties arise, due to the presence of the fractional nonlocal operator (−∆) s .In order to overcome these difficulties we need to perform a subtle analysis and to prove some regularity results for the eigenfunctions of (−∆) s (see Lemma 3.1) and for weak solutions of nonlocal problems (see Lemma 3.2).These complications are not only technical, but reflect the nonlocality of the problem.
The paper is organized as follows.Section 2 is devoted to the construction of the minimal and maximal curves of the Dancer-Fučík spectrum in an abstract setting introduced in [11] and to some abstract existence results of linking type got in [10].In Section 3 we introduce the functional setting, give the variational formulation of problem (1.1) and prove some regularity results for nonlocal problems.Finally in Section 4 and Section 5 we prove the main results of the paper, concerning the existence of a nontrivial solution for problem (1.1).

Abstract setting
This section is devoted to some preliminary results we need in order to prove our existence theorems for problem (1.1).In particular in Subsection 2.1 we recall some properties of the Dancer-Fučík spectrum in a suitable abstract setting, while Subsection 2.2 deals with some critical points theorems in presence of a suitable linking geometry, recently got in [10].

Dancer-Fucik spectrum in an abstract setting
First we briefly recall the construction of the minimal and maximal curves of the Dancer-Fučík spectrum in an abstract setting introduced in Perera and Schechter [11,Chapter 4].
Let H be a Hilbert space with the inner product (•, •) and the associated norm • .Recall that an operator ϕ : H → H is monotone if (ϕ(u) − ϕ(v), u − v) ≥ 0 for all u, v ∈ H and that ϕ ∈ C(H, H) is a potential operator if ϕ = Φ ′ for some functional Φ ∈ C 1 (H, R), called a potential for ϕ.Assume that there are positive homogeneous monotone potential operators p, n ∈ C(H, H) such that We use the suggestive notation u + = p(u), u − = −n(u), so that This implies Let A be a self-adjoint operator on H with the spectrum σ(A) ⊂ (0, ∞) and A −1 compact.Then the spectrum σ(A) of A consists of isolated eigenvalues λ l , l ≥ 1 of finite multiplicities satisfying 0 < λ ) is a Hilbert space with the inner product and the associated norm We have and the embedding is compact since A −1 is a compact operator.
Let E l be the eigenspace of λ l and N l and M l be defined as follows We assume that w ± = 0 for all w ∈ M 1 \ {0}.
The set Σ(A) consisting of points (a, b) ∈ R 2 for which the equation has a nontrivial solution is called the Dancer-Fučík spectrum of A. It is a closed subset of R 2 (see [11,Proposition 4.4.3]).Since equation (2.1) reduces to Au = λu when a = b = λ, Σ(A) contains the points (λ l , λ l ).
It is easily seen that A is a potential operator with the potential The potentials of p and n are respectively (see [11,Proposition 4.3.2]).So solutions of equation (2.1) coincide with critical points of the C 1 -functional I : D → R given by and (a, b) ∈ Σ(A) if and only if I(•, a, b) has a nontrivial critical point.Let , is strictly concave in v and strictly convex in w, i.e., for (see [11,Proposition 4.6.1]).
is the unique solution of Moreover, τ is continuous on N l ×Q l , τ (v, λ l , λ l ) = 0 for all v ∈ N l , and and and (ii) µ l is a continuous and strictly decreasing function, Thus, are strictly decreasing curves in Q l that belong to Σ(A).They both pass through the point (λ l , λ l ) and may coincide.The region } above the upper curve C l are free of Σ(A).They are the minimal and maximal curves of Σ(A) in Q l in this sense.Points in the region } between C l and C l , when it is nonempty, may or may not belong to Σ(A).

Abstract existence results
In this subsection we recall the abstract existence results for problems with jumping nonlinearities got by Perera and Sportelli in [10], that we will use to prove Theorems 1.1 and 1.2.Consider the equation where a, b > 0 and f ∈ C(D, H) is a potential operator.Let F ∈ C 1 (D, R) be the potential of f that satisfies F (0) = 0, i.e., ). Solutions of equation (2.7) coincide with critical points of the C 1 -functional E : D → R defined as follows where I is given in (2.2).We assume that where and there exists e ∈ D \ B with −e / ∈ B such that where Finally, we state the linking theorem from Perera and Sportelli [10] that we will use to prove Theorem 1.2.At this purpose, recall that a mapping ϕ : Y → Z between linear spaces is positive homogeneous if ϕ(tu) = tϕ(u) for all u ∈ Y and t ≥ 0 and denote by H the class of homeomorphisms h of a Banach space X onto itself such that h and h −1 map bounded sets into bounded sets.Theorem 2.4 ([10, Theorem 3.7]).Let E be a C 1 -functional defined on a Banach space X with norm • .Let X = N ⊕ M, with N finite dimensional and M closed and nontrivial.Let θ ∈ C(M, N) be a positive homogeneous map and let T : N → X be a bounded linear map.
If I N − T is sufficiently small, where I N is the identity map on N, and there exist ρ > 0 and e ∈ X \ T (N) such that where where Moreover, if E satisfies the (PS) c condition, then c is a critical value of E.

Functional setting and energy functional
In this section we introduce the functional setting and give the variational formulation of problem (1.1).We also prove some regularity results useful in the sequel and we recall some properties of the best fractional Sobolev constant. Let be the Gagliardo seminorm of a measurable function u : R N → R and let be the fractional Sobolev space endowed with the norm where |•| 2 denotes the norm in L 2 (R N ).We work in the closed linear subspace Problem (1.1) fits into the abstract setting of Subsection 2.1 with , and A equal to the inverse of the solution operator Since the embedding H s 0 (Ω) ֒→ L 2 (Ω) is compact by [16,Lemma 8] and [19,Lemma 9], For a complete study of the eigenvalues and eigenfunctions of the fractional Laplace operator (−∆) s (and its generalization) we refer to [13, Proposition 2.3], [18, Proposition 9 and Appendix A], [17,Proposition 4] and [20,Corollary 8].Let E l be the eigenspace of λ l , and let

Some regularity results
In this subsection we prove some regularity results useful in the sequel.
Then u ∈ L ∞ (R N ) and there exists C > 0, possibly depending on N, s and κ, such that Proof.We use arguments similar to the ones considered in [20, Proposition 9] and [17,Proposition 4].We may assume that u does not vanish identically.Let δ > 0, to be taken appropriately small in what follows (the choice of δ will be done on (3.15) below).Up to multiplying u by a small constant, we may and do assume that Moreover, the sequence w k satisfies the following properties: ) and Indeed, since C k+1 > C k , it is easy to see that v k+1 < v k a.e. in R N , which yields (3.7).
Finally, to get (3.9), it is enough to note that by (3.8) we obtain Now, we prove that for any k ∈ N For this, let x ∈ {w k+1 > 0} .Then u(x) − C k+1 > 0 and so, by the properties of C k , we have which gives (3.10).We also have that v k+1 (x) − v k+1 (y) = u(x) − u(y) for any x, y ∈ R N .From this, [20, Lemma 10], (3.2), the definition of w k+1 and the fact that u is a weak solution of (3.2), we deduce that (3.12) Now we use the Hölder inequality and the fractional Sobolev inequality to get that for some positive constant c depending only on N and s .Consequently, by (3.9), (3.12) and (3.13), we get that where C = (1 + c * ) 2 (4s/N +1) , with c * = c2 (4s/N +1) max{κ, g ∞ }, and . Now we are ready to perform our choice of δ: precisely, we assume that δ > 0 is so small that We also fix η ∈ δ γ−1 , , so that, since C > 1 and γ > 1, η ∈ (0, 1) .

Energy functional and best fractional critical Sobolev constant
In this subsection we define the energy functional associated with problem (1.1).Later we recall some results related with the best fractional critical Sobolev constant.
The variational functional E : H s 0 (Ω) → R associated with problem (1.1) is defined by where and the potential It is easily seen that F clearly satisfies (F 1 ) and (F 2 ).It also follows from standard arguments that the energy functional E satisfies (F 3 ) with where is the best fractional Sobolev constant.The infimum in (3.19) is attained on the functions where the constant c N, s > 0 is chosen so that (see Servadei and Valdinoci [19]).Fix In the sequel we will apply Theorems 2.3 and 2.4 taking e = u ε, µ with ε > 0 sufficiently small and µ ≥ µ 0 sufficiently large.We have the following estimates for u ε, µ 0 (see Servadei and Valdinoci [19,Propositions 21 & 22]): for some constants c 1 , . . ., c 4 > 0.
4 Proof of Theorem 1.1 This section is devoted to the proof of Theorem 1.1.At this purpose we will use the abstract linking result stated in Theorem 2.3.First of all, we note that, since u ± = (−u) ∓ , u solves (1.3) (resp.(1.1)) if and only if −u solves (1.3) (resp.(1.1)) with a and b interchanged.So Σ(−∆) is symmetric about the line a = b and we may assume without loss of generality that a ≤ b.
We start with some preliminary results.

Proof of Theorem 1.2
In this section we prove Theorem 1.2 using the abstract result stated in Theorem 2.4.
First of all, we provide some preliminary results.

. 4 )
Since I(u, a, b) is nonincreasing in a for fixed u and b, and in b for fixed u and a, n l−1 (a, b) and m l (a, b) are nonincreasing in a for fixed b, and in b for fixed a.Moreover, n l−1 and m l are continuous on Q l and n l−1 (λ l , λ l ) = 0 = m l (λ l , λ l ) (see[11, Lemma 4.7.6 & Proposition  4.7.7]).For a ∈ (λ l−1 , λ l+1 ), set

(F 3
) there exists c * > 0 such that for each c ∈ (0, c * ), every (PS) c sequence of E has a subsequence that converges weakly to a nontrivial critical point of E. Now we can state the following results.Theorem 2.3 ([10, Theorems 4.1 and 4.2]).Assume (F 1 )-(F 3 ), let (a, b) ∈ Q l , and let B = {v + τ (v, a, b) : v ∈ N l }, where τ is given in Theorem 2.1-(ii).Then equation (2.7) has a nontrivial solution in each of the following cases: (i) b < ν l−1 (a) and there exists e ∈ D \ N l−1 such that byServadei and Valdinoci [17, Proposition 4]  and taking into account that the eigenfunctions of (−∆) s vanish in R N \ Ω.Then N l ⊂ C s (R N ) by Ros-Oton and Serra [12, Proposition 1.1] and (3.1).Now iterating Ros-Oton and Serra[12, Proposition
see the proof of [11, Proposition 4.7.1]), then ũj k converges to v + τ (v, a, b) in H s 0 (Ω) and hence also in L 2 (Ω).This is a contradiction since (ũ j k ) is an orthonormal sequence in L 2 (Ω).Hence the claim is proved.Now, to verify (2.9), we use Lemma 4.2 with K = S ∩ B. As in the proof of [10, Theorem 4.2], (4.13) holds.It only remains to show that K ⊂ C 2 (Ω) ∩ C(Ω) and (4.2) holds.At this purpose let u ∈ K.By Theorem 2.1,