AMBROSETTI-PRODI TYPE RESULTS FOR DIRICHLET PROBLEMS OF FRACTIONAL LAPLACIAN-LIKE OPERATORS

We establish Ambrosetti–Prodi type results for viscosity and classical solutions of nonlinear Dirichlet problems for the fractional Laplace and comparable operators. In the choice of nonlinearities we consider semi-linear and super-linear growth cases separately. We develop a new technique using a functional integration-based approach, which is more robust in the non-local context than a purely analytic treatment.


Introduction and statement of results
In this paper our goal is to present a counterpart for the fractional Laplacian and operators comparable in a specific sense, of the classical Ambrosetti-Prodi problem studied for a class of elliptic differential operators with nonlinear terms. In contrast with topological and variational methods used in the classical context, we propose a new technique based on a path integration approach, which accommodates a large class of non-local operators going well beyond the fractional Laplacian, and also applies to viscosity solutions. This larger class is motivated by a number of applications including operators with Lévy jump kernels having a lighter than polynomial tail, however, an extension to this class requires a number of extra steps and concepts, which will be pursued in a future work. Apart from this, another general advantage of our approach seems to be that it is more robust than purely analytic techniques, dealing better with the difficulties resulting from boundary roughness. Our techniques and framework have been developed recently in [12,13], to which we now intend to add the new dimension of including nonlinearities. First we briefly recall the original problem, then state our results, and in the next section present the proofs.
Let D ⊂ R d be a bounded open domain with a C 2,α (D) boundary, α ∈ (0, 1), and consider the Dirichlet problem ∆u + f (u) = g(x) in D , u = 0 on ∂D, (1.1) (1) there is a closed connected manifold M 1 ⊂ C 0,α (D) of codimension 1, with the property that there exist M 0 , M 2 such that C 0,α (D) \ M 1 = M 0 ⊔ M 2 , (2) the Dirichlet problem (1.1) has no solution if g ∈ M 0 , has a unique solution if g ∈ M 1 , and has exactly two solutions if g ∈ M 2 .
The problem formulates in the wider context of invertibility of differentiable maps between Banach spaces, in fact, M 1 is the set of elements u on which the Fréchet derivative of L is not locally invertible. Also, as it is seen from condition (1.2), this split behaviour shows that the existence and multiplicity of solutions is conditioned by the crossing of the nonlinear term with the principal eigenvalue of the linear part. Following this fundamental observation, much work has been done in the direction of relaxing the conditions or generalizing to further non-linear partial differential equations or systems. A first contribution has been made by Berger and Podolak proposing a useful reformulation of the problem. Write where ϕ 1 is the principal eigenfunction of the Dirichlet Laplacian, h is in the orthogonal complement of ϕ 1 , L 2 -normalized to 1, and ρ ∈ R, so that (1.1) becomes In [6] it is then shown that there exists ρ * (h) ∈ R, continuously dependent on h, such that for ρ > ρ * (h) the equivalent Dirichlet problem has no solution, for ρ = ρ * (h) it has a unique solution, and for ρ < ρ * (h) it has exactly two solutions. For further early developments we refer to the works of Kazdan and Warner [24] relaxing the assumptions, Dancer [16], Amann and Hess [1] identifying a suitable growth condition on f 1 , and Ruf and Srikanth [32] turning to the super-linear case. More recent papers exploring different perspectives include [3,15,18,20,21,27,35,36], and for useful surveys we refer to de Figueiredo [19] and Mahwin [26]. For non-local Hamilton-Jacobi equations see [17], and for systems of non-local equations [28]. Let D ⊂ R d be a bounded domain with C 2 boundary, s ∈ (0, 1), and consider the fractional Laplacian-like operator where k : S d−1 → (0, ∞) is a symmetric (i.e., k(z) = k(−z)), Borel-measurable function satisfying the non-degeneracy condition As it can be seen from Lemma 2.9 below, our proof techniques use fine boundary behaviour of the solutions of the Dirichlet problem. It is known from [30] that such a behaviour may not hold for a general non-degenerate kernel k defined on R d . This is the reason why we restrict ourselves to the kernel functions of above type. Motivated by the problem (1.3), in this paper we are interested in the existence and multiplicity of solutions of where Φ 1 is the Dirichlet principal eigenfunction of L in D, ρ ∈ R, and h ∈ C α (D) for some α > 0. We also assume that Φ 1 ∞ = 1. Below we will consider viscosity solutions, however, we will also discuss a sufficient condition on f so that every viscosity solution becomes a classical solution.
Let V ∈ C(D), which will be referred to as a potential. We use the notation C b,+ (R d ) for the space of non-negative bounded continuous functions on R d . Also, we denote by C 2s+ (D) the space of continuous functions on D with the property that if ψ ∈ C 2s+ (D), then for every compact subset K ⊂ D there exists γ > 0 with f ∈ C 2s+γ (K). Define The principal eigenvalue of −L + V is defined as (1.4) For easing the notation, we will simply write λ * for the above. This widely used characterization of the principal eigenvalue originates from the seminal work of Berestycki, Nirenberg and Varadhan [4]. Descriptions in a similar spirit for a different class of non-local Schrödinger operators have been obtained in [5], while in [11,17,33] non-local Pucci operators have been considered. Recently, we proposed in [13] a probabilistic approach using a Feynman-Kac representation to establish characterizations of the principal eigenvalue and the corresponding semigroup solutions. Our first result concerns the existence of the principal eigenfunction and of a solution of the Dirichlet problem.
We will also need the following refined weak maximum principle for viscosity solutions.
Remark 1.1. On completion of this paper we have learnt that [17] obtained results similar to Theorems 1.1-1.2, using a different technique than ours. In our understanding, these and related methods in the literature, applied also for other purposes, depend on the comparability of the used non-local operators with the Riesz kernel and the fractional Laplacian. We emphasize that in this paper we develop a path integration-based approach which, like in the framework first set in [11,12,13], is applicable for a large class of non-local operators (Markov generators of Lévy processes) without a similar restriction, also covering qualitatively different jump kernels. Since this will need further probabilistic machinery, it will be presented elsewhere, and we limit ourselves to the fractional Laplacian here. A recent paper [9] deals with a class of non-local operators and extends some of results of this paper using a probabilistic framework.
Next we impose the following Ambrosetti-Prodi type condition on f .
(1) f is Hölder continuous in x, locally with respect to u, and locally Lipschitz continuous in u, uniformly in x ∈D, (2) there exist V 1 , V 2 ∈ C α (D), for some α > 0, such that (3) f has at most linear growth, i.e., there exists a constant C > 0 such that for all (x, q) ∈D × R, or (3') L = −(−∆) s (i.e. k is constant), d > 1 + 2s and there exists a positive continuous function a 0 such that where the above limit holds uniformly in x ∈D.
When referring to Assumption [AP] below, we will understand that conditions (1), (2) and one of (3) or (3') hold. In what follows, we assume with no loss of generality that f (x, 0) = 0, otherwise h can be replaced by h − f (·, 0). Now we are ready to state our main result on the fractional Ambrosetti-Prodi problem.
To prove our main Theorem 1.3, like in classic proofs such as in [19,20], in our context too the viscosity characterization of the principal eigenfunction plays a key role. In Theorem 1.1 therefore first we obtain such a characterization. The refined maximum principle shown in Theorem 1.2 will also be a key object towards the proof of the fractional Ambrosetti-Prodi phenomenon. We will rely on our recent work [13], in which we proposed a method based on Feynman-Kac representations to establish Aleksandrov-Bakelman-Pucci (ABP) estimates for semigroup solutions of non-local Dirichlet problems for a large class of operators, including but going well beyond the fractional Laplacian. We will also show that every classical solution in our context here is also a semigroup solution and thus a generalized ABP estimate can be established for these solutions, which will be essential for obtaining the a priori estimates.

Preliminaries
We begin by recalling some notations and results from [12,13], which will be used below. Let (Ω, F, P) be a complete probability space, and (X t ) t≥0 be an isotropic Lévy process on this space with infinitesimal generator L. Given a function V ∈ C(D), called potential, the corresponding Feynman-Kac semigroup is given by , t > 0, is a Hilbert-Schmidt operator on L 2 (D) with continuous integral kernel in (0, ∞)×D×D. Moreover, every operator T t has the same purely discrete spectrum, independent of t, whose lowest eigenvalue is the principal eigenvalue λ * having multiplicity one, and the corresponding principal eigenfunction Ψ ∈ L 2 (D) is strictly positive. We also have from [12,Lem. 3.1] that Ψ ∈ C 0 (D), where C 0 (D) denotes the class of continuous functions on R d vanishing in D c . Since Ψ is an eigenfunction in semigroup sense, we have for all t > 0 that Let (D n ) n∈N be a collection of strictly decreasing domains with the property that ∩ n≥1 D n = D, and each D n having its boundary satisfying the exterior cone condition. Denote by λ * n the principal eigenvalue in sense of (1.4). The following result will be useful below (see also, [13,Lem. 4
V and λ * V denote the principal eigenvalues corresponding to the potentials V andṼ , respectively.
Proof. Existence of a unique principal eigenfunction follows from [33, Th. 1.1] (see also [17]). Note that [33] considers the case s > 1 2 due to the presence of the drift term and the same proof would go through in our setting. Using [34, Th. 1.3] we can show that the eigenfunction belongs to C 2s+ (D). Then the strict monotonicity of the eigenvalue with respect to domains follows from [33,Theorem 5.1]. Using the arguments of [11, Th. 1.6] it can be shown that lim n→∞ λ * n = λ * . Part (2) again follows from [33, Th. 5.1].
Since the theory developed in [12] is probabilistic while here we are concentrating on viscosity solutions, we point out the relationship between these notions of solution (compare also with [13, Semigroup super-solutions are defined in an analogous way. Lemma 2.1. Suppose that V, g ∈ C(D), and let u satisfy We have the following: is a semigroup sub-solution (resp., super-solution) of (2.2), then it is also a viscosity sub-solution (resp., super-solution).
is a classical sub-solution (super-solution) of (2.2), then it is also a semigroup sub-solution (resp., super-solution).
Proof. Consider part (1). Choose a point x ∈ D, and let ϕ ∈ C 2 (D) be a test function that (strictly) touches u at x from above, i.e., for a ball , ϕ r,n ≥ u, and ϕ r,n → ϕ r almost surely, as n → ∞. Since u is a semigroup subsolution, we have that e − s 0 V (Xp)dp g(X s ) ds, is a submartingale with respect to the natural filtration of (X t∧τ D ) t≥0 , see also [13], hence by optional sampling we obtain that where τ r denotes the first exit time from the ball B r (x). On the other hand, by applying Itô's formula on ϕ r,n we obtain On dividing both sides by t and letting t → 0, it follows that Thus by letting n → ∞, we obtain which proves the first part of the claim. Next consider part (2). By the property of u we note that Lu is continuous in D. Consider a sequence of open sets K n ⋐ K n+1 ⋐ D and Applying Itô's formula to ψ m , we get that where τ n denotes the first exit time from the set K n . First letting m → ∞ and then n → ∞ above, and using the fact that τ n ↑ τ D almost surely, we obtain This shows that u is a semigroup subsolution.

Proof of Theorem 1.1
Now we are ready to prove our first theorem.
Proof. First consider (a). As discussed in Proposition 2.1, there exists an eigenpair (λ * , Next we consider (b). The main idea in proving (1.6) is to use Schauder's fixed point theorem. Consider a map T : C 0 (D) → C 0 (D) defined such that for every ψ ∈ C 0 (D), T ψ = ϕ is the unique viscosity solution of − Lϕ = g − V ψ in D, and ϕ = 0 in D c .
for a constant c 1 = c 1 (D, d, s). This implies that T is a compact linear operator. It is also easy to see that T is continuous.
In a next step we show that the set for some µ ∈ [0, 1]. As argued above, we note that ϕ ∈ C 2s+ (D) ∩ C 0 (D Let λ * 0 > 0 be the principal eigenvalue corresponding to the potential V = 0. Then from the concavity of the map µ → λ * µV , which results from (2.8) by applying Young's inequality, it follows that λ * µV ≥ λ * V ∧ λ * 0 = 2δ > 0. Hence by using (2.8) and the continuity of µ → λ * µV , we find constants c 3 > 0, µ 0 > 1, such that for every µ ∈ [0, µ 0 ] we have Since ϕ is a semigroup solution, we have that Letting t → ∞, using (2.9) and Hölder inequality, it is easily seen that the first term at the right hand side of the above vanishes. Again by (2.9), we have for x ∈ D

Proof of Theorem 1.2
First we show that the C 2 -class of test functions can be replaced by functions of class C 2s+ in the definition of the viscosity solution.
Suppose that there exists an open set N ⋐ D, containing x, and a function ϕ ∈ C 2s+ (N) satisfying ϕ(x) = u(x) and ϕ > u in N \ {x}. Define Proof. Consider a sequence of functions in (ϕ m ) m∈N , C 2 in a neighbourhood of x, and such that ϕ m − ϕ C 2s+α (N ) → 0, for some α > 0, as m → ∞. This is possible since ϕ ∈ C 2s+ (N ). Let By the definition of the viscosity subsolution we find where C(d, s) is the normalizing constant for fractional Laplacian and r > 0 is chosen to satisfy B 2r (x) ⋐ N . It is easily seen that we can let m → ∞ above and use the continuity of V, g, u to obtain −Lϕ N (x) + V (x)u(x) ≤ g(x), which shows the claim.
Next we prove our second theorem stated in the previous section.
Proof of Theorem 1.2. Let w = u − v. By [14, Th. 5.9] it then follows that in viscosity sense. Note that w ≤ 0 in D c , while we need to show that w ≤ 0 in R d . Suppose, to the contrary, that w + > 0 in D. Using Proposition 2.1, we find a domain D 1 ⋑ D with a C 1 -boundary and λ * 1 > 0, where λ * 1 is the principal eigenvalue for D 1 and potential V . In fact, we may take V as a C α -extension from D to D 1 . Let Ψ 1 ∈ C 2s+ (D 1 ) ∩ C 0 (D 1 ) be the corresponding positive principal eigenfunction. Thus we have Since min D Ψ 1 > 0, it follows that c 0 is finite, and w + > 0 implies that c 0 > 0. Then Φ = c 0 Ψ 1 − w necessarily vanishes at some point, say x 0 ∈ D. This follows from the fact that w + = 0 on ∂D.

Proof of Theorem 1.3
Now we turn to proving our main result on the fractional Ambrosetti-Prodi phenomenon. The strategy of proof will be divided in the following steps.
(1) First we find ρ 1 such that for every ρ ≤ ρ 1 there exists a minimal solution of (P ρ ). This will be done in Lemmas 2.3 and 2.4 below. (2) Next we find ρ 2 > ρ 1 such that no solution of (P ρ ) above ρ 2 exists. This is the content of Lemma 2.7 and Lemma 2.8. (3) Finally, we follow the arguments in [19] to find the bifurcation point ρ * . We begin by showing the existence of a sub/super-solution, which will be used for constructing a minimal solution.
(2.17) Moreover, the above u can be chosen to be minimal in the sense that ifũ ∈ C 2s+ (D) ∩ C 0 (D) is another solution of (2.17), thenũ ≥ u in R d .
Proof. The proof is based on the standard monotone iteration method. Denote m = minD u and M = maxDū. Let θ > 0 be a Lipschitz constant for f (x, ·) on the interval [m, M ], i.e., . Consider the solutions of the following family of problems: for some constants α, κ 1 . Hence there exists u ∈ C α (R d ) ∩ C 0 (D) such that u (n) → u in C 0 (D) as n → ∞. Using the stability of viscosity solutions, it then follows that u is a viscosity solution to We can now apply the regularity estimates from [34] to show that u ∈ C 2s+ (D).
To establish minimality we consider a solutionũ of (2.17) in C 2s+ (D)∩C 0 (D). From Lemma 2.3(3) we have u ≤ũ in R d . Thusū can be replaced byũ, and the above argument shows that u ≤ũ. Now we derive a priori bounds on the solutions of (P ρ ). Our first result bounds the negative part of solutions u of (P ρ ). We recall that under the standing assumptions on f , any viscosity solution of (P ρ ) is an element of C 2s+ (D) ∩ C 0 (D), and thus also a classical solution.
Proof. First observe that if u is a solution to (P ρ ) for some ρ ≥ −ρ, then Defining w = u ∧ 0 we see that w is a viscosity super-solution of the above equation, i.e.,
Our next result provides a lower bound on the growth of the solution for large ρ.
One may notice that we have not used the second condition in (1.7) so far. The next result makes use of this condition to establish an upper bound on the growth of u.
Proof. Suppose, to the contrary, that there exists a sequence (ρ n , u n ) n∈N satisfying (P ρ ) with ρ n ≥ −ρ and u n ∞ → ∞. From Lemma 2.5 it follows that u + n ∞ = u n ∞ . Define v n = un un ∞ . Then Since H n ∞ is uniformly bounded by Lemmas 2.5-2.6, it follows by [11,Th. 2.6] that for some α > 0. Hence we can extract a subsequence of (v n ) n∈N , denoted by the original sequence, such that it converges to a continuous function v ∈ C 0 (D) in C(R d ). Denote Then using (1.9) and (2.20), we get −Lv n − V 2 v n ≥ G n in D.
Using Lemma 2.1 we have that v n is a semigroup super-solution, i.e., for every Letting n → ∞ in (2.21) and using the uniform convergence of G n and v n , we obtain Since v ∞ = 1 and v ≥ 0 in R d , it is easily seen from (2.22) that v > 0 in D. Again, by Lemma 2.1, we see that −Lv − V 2 v ≥ 0 in D, in viscosity sense. Hence it follows that λ * ((−∆) s − V 2 ) ≥ 0, contradicting (1.7). This proves the first part of the result. The second part follows by Lemma 2.6 and (2.19).
The following result will be useful for tackling the super-linear case. We note that the general idea of the a priori bound below has its origins in the work of Gidas and Spruck [22], see also the more recent [7] for a non-local version. Lemma 2.8. Let Assumption [AP](3') hold. Then for everyρ > 0 there exists C 0 such that for every solution u of (P ρ ), with ρ ≥ −ρ we have (2.23) In particular, there exists ρ 2 > 0 such that (P ρ ) does not have any solution for ρ ≥ ρ 2 .
Proof. First we establish (2.23) for all ρ ≥ 1. Suppose, to the contrary, that there exists (u n , ρ n ) n∈N , ρ n ≥ 1, satisfying (P ρ ) with the property that u n ∞ ≥ nρ and θ n = dist(x n , ∂D). Using compactness, we may also assume that x n → x 0 ∈D as n → ∞. We split the proof into two cases. Case 1. Suppose that lim sup n→∞ θn γn = +∞. Define w n (x) = 1 un ∞ u n (γ n x + x n ). We then have (2.26) We choose a subsequence, denoted is the same way, such that lim n→∞ θn γn = +∞. Then for any given k ∈ N there is a large enough n 0 satisfying B k (0) ⊂ 1 γn (D − x n ) for all n ≥ n 0 . Therefore, the right hand side of (2.26) is uniformly bounded in B k (0). Since w n = w n (0) = 1, it follows that for some α > 0, w n C α (B k/2 (0)) is bounded uniformly in n (see [14]). Thus we can extract a subsequence (w n ) n∈N such that w n → w ∈ C b,+ (R d ) locally uniformly. Hence, by the stability of viscosity solutions (−∆) s w = a 0 (x 0 )w p in R d , w(0) = 1. By the strong maximum principle we also have w > 0. However, no such solution can exist due to the Liouville theorem [29,Th. 1.2], and hence we have a contradiction in this case.
. Then the estimate follows from the local Lipschitz property of f and Lemma 2.5. Hence, using (P ρ ) we obtain −(−∆) s u n + κ 1 (1 + u n p−1 ∞ )sgn(u n )u n ≥ −ρ n − h ∞ in D. Denote by C n = ρ n + h ∞ . Applying Lemma 2.1 we get that for t ≥ 0, It follows from the proof of [8, Th. 1.1] that there exist constants κ 2 and η ∈ (0, 1), not depending on x n , such that for t = κ 2 θ 2s n we have P xn (τ D > t) ≤ η.
Inserting this choice of t in the above expression we obtain Thus by the assertion and (2.25) it follows that for all large n we have This gives (2.27), since θ n → 0.
Hence we may assume that, up to a subsequence, holds. Then using again an argument similar to above, we obtain a positive bounded solution Note that h ∞ is uniformly bounded for ρ ∈ [−ρ, 1]. Then (2.24) follows from the previous argument. The other claim follows by (2.23) and Lemma 2.6.
With the above results in hand, we can now proceed to prove Theorem 1.3. Define A = ρ ∈ R : (P ρ ) has a viscosity solution .
Using Lemma 2.3(3) and from the proof of Lemma 2.4 we have a minimal solution of (P ρ ) with ρ = ρ ′ . Next we show that there are at least two solutions for ρ < ρ * .
Recall that d :D → [0, ∞) is the distance function from the boundary of D. We can assume that d is a positive C 1 -function in D. For a sufficiently small ε > 0, to be chosen later, consider the Banach space In fact, it is sufficient to consider any ε strictly smaller than the parameter α < s ∧ (1 − s) in [31, Th. 1.2]. Since d s is s-Hölder continuous inD, it is routine to check that X ⊂ C ε (D). For ρ ∈ R and m ≥ 0 we define a map K ρ : X → X as follows. For v ∈ X, K ρ v = u is the unique viscosity solution (see  Proof. We borrow some of the arguments of [19] with a suitable modification. Pickρ ∈ (ρ, ρ * ) and letū be a solution of (P ρ ) with ρ =ρ. It then follows that The results in [23] are proved for (−∆) s (i.e. k constant), but the similar argument works with the barrier function constructed in [30,Lem. 3.4] giving us Hopf's lemma in our setting. Similarly, we can compare also u andū. We define m to be a Lipschitz constant of f (x, ·) in the interval [min u, maxū]. Also, definẽ f (x, q) = f (x, (u(x) ∨ q) ∧ū(x)) + m(u(x) ∨ q) ∧ū(x).
Note that f is bounded and Lipschitz continuous in q, and also non-decreasing in q. We define another mapK ρ : X → X as follows: for v ∈ X,K ρ v = u is the unique viscosity solution of − Lu + mu =f (x, v) + ρΦ + h in D, and u = 0 in D c . (2.28) It is easy to check that K ρ is a compact mapping. Using again [30, Th. 1.2], we find r satisfying sup K ρ v X : v ∈ X < r.
We fix this choice of r. Using the regularity estimate of [34], we see that the solution u in (2.28) is in C 2s+ (D). Therefore, The other estimates can be obtained similarly. Finally, this implies thatK ρ v ∈ O, for all v ∈ X. Moreover, 0 / ∈ (I −K ρ )(∂D). Then by the homotopy invariance property of degree we find that deg(I −K ρ , O, 0) = 1. SinceK ρ coincides with K ρ in O, we obtain deg(I − K ρ , O, 0) = 1.
Similarly as before, define S ρ : X → X such that for v ∈ X, u = S ρ v is given by the unique solution of −Lu = f (x, v) + ρΦ 1 + h(x) in D, and u = 0 in D c . Then the standard homotopy invariance of degree gives that deg(I −S ρ , O, 0) = 1. This observation will be helpful in concluding the proof below.
Proof of Theorem 1.3. Using Lemma 2.9 we can now complete the proof by using [19,20]. Recall the map S ρ defined above, and fix ρ < ρ * . Denote by O R a ball of radius R in X. From Lemmas 2.7 and 2.8 we find that deg(I − Sρ, O R , 0) = 0 for all R > 0,ρ ≥ ρ 2 . Using again Lemmas 2.7, 2.8 and [31, Th. 1.2], we obtain that for everyρ there exists a constant R such that u X < R for each solution u of (P ρ ) withρ ≥ −ρ. Fixingρ > |ρ| and the corresponding choice of R, it then follows from homotopy invariance that deg(I − S ρ , O R , 0) = 0. We can choose R large enough so that O ⊂ O R . Since deg(I − S ρ , O, 0) = 1, as seen above, using the excision property we conclude that there exists a solution of (P ρ ) in O R \ O. Hence for every ρ < ρ * there exist at least two solutions of (P ρ ). The existence of a solution at ρ = ρ * follows from the a priori estimates in Lemmas 2.7 and 2.8, the estimate in [11,Th. 2.6], and the stability property of the viscosity solutions. This completes the proof of Theorem 1.3.