Three nontrivial solutions for nonlinear fractional Laplacian equations

We study a Dirichlet-type boundary value problem for a pseudodifferential equation driven by the fractional Laplacian, proving the existence of three nonzero solutions. When the reaction term is sublinear at infinity, we apply the second deformation theorem and spectral theory. When the reaction term is superlinear at infinity, we apply the mountain pass theorem and Morse theory.


Introduction
The present paper deals with the following Dirichlet-type boundary value problem for a nonlinear equation driven by the fractional Laplacian: where Ω ⊆ R N (N > 1) is a bounded domain with a C 2 boundary, Ω c = R N \ Ω, s ∈ (0, 1), and f : Ω × R → R is a Carathéodory function. The fractional Laplacian operator is defined for any sufficiently smooth function u : R N → R and all x ∈ R by where C N,s > 0 is a suitable normalization constant. Throughout the paper we will always assume C N,s = 1 (for a precise evaluation of C N,s , consistent with alternative definitions of the fractional Laplacian, see [10,Remark 3.11]). Fractional operators have gained increasing popularity in recent years. This is due both to the intrinsic mathematical interest of such subject, and to the various applications that they allow. Indeed, nonlocal pseudodifferential operators such as (−∆) s are naturally involved in continuum mechanics, population dynamics, game theory and other phenomena, as the infinitesimal generators of Lévy-type stochastical processes (see [12]). Roughly speaking, the outstanding feature of operators like (−∆) s is nonlocality, i.e., the dependence of (−∆) s u(x) on the values of u(y) not only for y conveniently near to x, but but for all y ∈ R N . While such nonlocality makes our operator particularly suitable to describe phenomena allowing 'jumps', it makes things delicate in dealing with regularity, sign, and other typically local attributes of solutions. This is one reason why the study of nonlinear equations involving (−∆) s (or closely related operators) started with the case when the domain is R N , providing existence of solutions, regularity, a priori bounds and maximum principles (see [10,11], and [3] for some existence results). The natural functional setting for such study is provided by fractional Sobolev spaces (see [16]). On the other hand, nonlocality obviously produces some difficulties in finding an analogous to Dirichlet-type boundary conditions on bounded domains. The standard formulation of the Dirichlet problem for fractional equations in a bounded domain Ω was set in the series of papers [34][35][36], simply by requiring that the solution u vanishes a.e. outside Ω. Our problem (1.1) follows such standard. While interior regularity of solutions of (1.1) can be handled just as in the unbounded case, boundary regularity and behaviour of solutions (e.g. the Hopf property) came forth as a serious difficulty, which was mostly overcome by means of weighted Hölder-type function spaces (see [5,21,23,33]). Once provided with the appropriate functional formulation, problem (1.1) becomes variational, in the sense that its weak solutions can be detected as critical points of a C 1 energy functional ϕ, defined on a fractional Sobolev space. So, we can prove existence and multiplicity of such solutions by applying to ϕ several abstract results of critical point theory, such as minimax principles (see [32]) and Morse theory (see [13]). Some results of this type can be found, for instance, in [6,14,17,25,27,29,38].
In the present paper, we will employ much of the research accomplished so far in order to prove the existence of three nonzero solutions for problem (1.1) (one positive, one negative, and the third with indefinite sign), when f (x, ·) has a subcritical growth and satisfies convenient conditions at zero and at infinity. Precisely, we will consider two cases: is sublinear at infinity, and at most linear at zero, then we apply the second deformation theorem and some spectral properties of (−∆) s (namely, a characterization of the second eigenvalue which, for the local case, goes back to [15]); is superlinear at infinity, and satisfies a mild version of the Ambrosetti-Rabinowitz condition, then we apply the mountain pass theorem and the Poincaré-Hopf identity based on the computation of critical groups (thus proving a nonlocal analogous of the result of [37]). In both cases, truncations of the energy functional ϕ will be an essential tool, so we will make use of a topological result established in [23], which relates local minimizers of the truncated and uncut functionals, respectively. Our work strongly relies on the joint application of mutually independent results, and we decided to privilege simplicity rather than generality. One possible generalization of our results is towards linear, nonlocal operators of the type where K : R N × R N → R + is a weight function exhibiting an asymptotic behaviour similar to that of the standard weight |x − y| N +2s (see [34]). Another possible extension may deal with the fractional p-Laplacian, namely the nonlinear, nonlocal operator defined by where p ∈ (1, ∞). Some existence and multiplicity results for fractional p-Laplacian problems, obtained through critical point theory and Morse theory, can be found in [22]. Nevertheless, the methods used in the present paper cannot be easily extended to (−∆) s p due to the lack of a complete boundary regularity theory like that developed in [33] for (−∆) s (some results in this direction are proved in [24]). The paper has the following structure: in Section 2 we recall the variational formulation of our problem and some basic properties of solutions, together with some results from critical point theory; in Section 3 we prove our multiplicity result for the sublinear case; and in Section 4 we deal with the superlinear case.

Preliminary results
In this section we recall some results that will be used in our arguments.
2.1. Variational formulation and some properties of problem (1.1). For all measurable function u : then we define the fractional Sobolev space (see [16]). We restrict ourselves to the subspace x ∈ Ω c }, which is a separable Hilbert space under the norm u = [u] s,2 (see [34]). We denote by H −s (Ω) the topological dual of H s 0 (Ω) and by ·, · the scalar product of H s 0 (Ω) (or the duality pairing between H −s (Ω) and H s 0 (Ω)). In this connection we mention the following useful inequality, holding for all u ∈ H s 0 (Ω): where u − stands for the negative part of u (see [23]). The critical exponent is defined as 2 * s = 2N N −2s , and the embedding H s 0 (Ω) ֒→ L p (Ω) is continuous and compact for all p ∈ [1, 2 * s ) (see [16,Lemma 8]). Moreover, we introduce the positive order cone x ∈ Ω}, which has an empty interior with respect to the H s 0 (Ω)-topology. The space H s 0 (Ω) provides the natural framework for the study of problem (1.1): In all the forthcoming results we will assume the following hypothesis on the nonlinearity f : |f (x, t)| a 0 (1 + |t| p−1 ) for a.e. x ∈ Ω and all t ∈ R (a 0 > 0, p ∈ (1, 2 * s )). Under such assumption, we are able to extend to problem (1.1) some basic results holding for elliptic boundary value problems, starting with a simple a priori bound: (Ω) weak solution of (1.1) one has u ∈ L ∞ (Ω) and u ∞ M ( u 2 * s ). While solutions of fractional equations exhibit good interior regularity properties, they may have a singular behaviour on the boundary. So, instead of the usual space C 1 (Ω), they are better embedded in the following weighted Hölder-type spaces. Set δ(x) = dist(x, Ω c ) for all x ∈ R N and define u δ s ∈ C α (Ω) (α ∈ (0, 1)), endowed with the norms respectively. For all 0 α < β < 1 the embedding C β δ (Ω) ֒→ C α δ (Ω) is continuous and compact. In this case, the positive cone C 0 δ (Ω) + has a nonempty interior given by From Proposition 2.2 and [33, Theorem 1.2] we have the following global regularity result: Let H 0 hold. Then there exist α ∈ (0, min{s, 1 − s}) and C > 0 s.t. for all u ∈ H s 0 (Ω) weak solution of (1.1) one has u ∈ C α δ (Ω) and u α,δ C(1 + u 2 * s ). We turn now to sign properties of solutions of (1.1). We begin with a weak maximum principle: is a solution of (1.1), then u is lower semicontinuous and u(x) 0 for all x ∈ Ω. Moreover, we have the following fractional Hopf lemma: . Remark 2.6. In its original version from [21], the above Hopf lemma requires that u satisfies (−∆) s u = f (x, u) pointwisely in Ω, while we deal with weak solutions. In fact, any weak solution u of (1.1) has a higher interior regularity than that displayed in Proposition 2.3, as u ∈ C 1,β (Ω) for any β ∈ (max{0, 2s − 1}, 2s) (see [33,Corollary 5.6]). Hence, also recalling that u = 0 in Ω c , one can see that the limit in (1.2) exists in R and the equation is satisfied pointwisely (see [24,Proposition 2.12]).

Now we introduce an energy functional for
By the continuous embedding H s 0 (Ω) ֒→ L p (Ω) we have ϕ ∈ C 1 (H s 0 (Ω)), and for all u, v ∈ H s 0 (Ω) So, recalling Definition 2.1, u is a solution of (1.1) iff ϕ ′ (u) = 0 in H −s (Ω). Among critical points of ϕ, local minimizers play a preeminent role. We recall, in this connection, a useful topological result relating such minimizers in the H s 0 (Ω)-topology and in C 0 δ (Ω)-topology, respectively (a fractional version of the classical result of [9]): Proposition 2.7. [23, Theorem 1.1] Let H 0 hold, ϕ be defined as above, and u ∈ H s 0 (Ω). Then, the following conditions are equivalent: In the proof of our result we will need some spectral properties of (−∆) s . Let us consider the following eigenvalue problem: Just as in the local case, we say that λ > 0 is an eigenvalue of (−∆) s if problem (2.4) has a nonzero solution u ∈ H s 0 (Ω), which is called a λ-eigenfunction. From the current literature we have rather complete information about the first two eigenvalues of (−∆) s : Proposition 2.8. The spectrum of (−∆) s consists of a nondecressing sequence 0 < λ 1 s (Ω) < λ 2 s (Ω) . . . of positive numbers, in particular: Note that (ii) above is a fractional version of a classical result of [15], and that Proposition 2.8 holds as well for (−∆) s p (see [7,19]). For further information about the spectra of (−∆) s and (−∆) s p see also [26,30,36].
2.2. Some recalls of critical point theory. Variational methods are based on abstract critical point theory, and the latter includes many results, depicting the rich topology that nonlinear and nonconvex functionals may exhibit. We recall here some well-known results which will be our major tools, mainly following [28] (see also [32]). Let (X, · ) be a reflexive Banach space, (X * , · * ) be its topological dual, and ϕ ∈ C 1 (X) be a functional. By K(ϕ) we denote the set of all critical points of ϕ, i.e., those ponts u ∈ X s.t.
Most results require the following Cerami compactness condition (a weaker version of the Palais-Smale condition): We recall a version of the mountain pass theorem (see [2,31] for the original result): Then, c η r and K c (ϕ) = ∅.
We will also use the second deformation theorem: Then, there exists a continuous deformation h : In particular, (i) -(ii) above mean that ϕ a is a strong deformation retract of ϕ b (see [28,Definition 5.33 (b)]). We conclude this section by recalling some basic notions from Morse theory (see [4,13] for details).
Let ϕ ∈ C 1 (X) satisfy (C) and u ∈ K c (ϕ) (c ∈ R) be an isolated critical point of ϕ, i.e., there exists Then, for all integer k 0 the k-th critical group of ϕ at u is defined as where H k (·, ·) is the k-th (singular) homology group of a topological pair (see [28,Definition 6.9]). All these groups are real linear spaces. Note that, by the excision property of homology groups, (2.5) is invariant with respect to U . In particular, if u ∈ K(ϕ) is a strict local minimizer and an isolated critical point, then for all k 0 we have

Now assume that inf
Then we can as well define the k-th critical group of ϕ at infinity as with c <c (this definition also is invariant with respect to c). Critical groups at critical points and at infinity are related by the Poincaré-Hopf formula (one of the Morse relations): Theorem 2.12. [28, Remark 6.58] Let ϕ ∈ C 1 (X) satisfy (C), a < b be real numbers s.t. the set Notation. Throughout the paper, B r (x) will denote the open ball of radius r > 0 centered at x ∈ R N , and C > 0 will be a constant whose value may change from line to line.
Then, f satisfies hypotheses H 1 .

The superlinear case
In this section we prove the existence of three non-zero solutions of problem (1.1) when f (x, ·) is superlinear at infinity. Following an idea first appeared in [37], we will apply the mountain pass theorem and Morse theory. Precisely, we make on the nonlinearity f the following assumtpions: x ∈ Ω and all t ∈ R (a 0 > 0, p ∈ (2, 2 * s )); (ii) f (x, t)t 0 for a.e. x ∈ Ω and all t ∈ [−σ, σ] (σ > 0); (iii) f (x, t)t −c 0 t 2 for a.e. x ∈ Ω and all t ∈ R (c 0 > 0); Condition H 2 (v) is a mild version of the classical Ambrosetti-Rabinowitz condition (see [32]), and since p < 2 * s we can always assume q < p in it.
. Then, f satisfies hypotheses H 2 with convenient a 0 , c 0 , and σ. This choice of f belongs in the class of concave-convex nonlinearities, whose study (in the classical case s = 1) started with [1].
By hypothesis H 2 (ii), problem (1.1) admits the zero solution. We focus now on constant sign solutions: Let H 2 hold. Then (1.1) admits at least two non-zero solutions u ± ∈ ±int (C 0 δ (Ω) + ). Proof. We define ϕ, ϕ ± as in (2.3), (3.1). We focus mainly on ϕ + . First we prove that ϕ + satisfies (C). Let (u n ) be a sequence in H s 0 (Ω) s.t. |ϕ + (u n )| C for all n ∈ N and (1 + u n )ϕ ′ + (u n ) → 0 in H −s (Ω). Then we have for all n ∈ N − u n 2 + Ω f + (x, u n )u n dx C, uniformly for a.e. x ∈ Ω. So we can find β, M > 0 s.t. f + (x, t)t − 2F + (x, t) βt q for a.e. x ∈ Ω and all t > M . We claim that (u n ) is bounded in L q (Ω). Indeed, for all n ∈ N we have u n q q = u + n q q + u − n q q . By the previous inequality we have and the latter is bounded by (4.1). Besides, using (2.1), we easily have ϕ ′ + (u n ) * u − n , and the latter tends to 0 as n → ∞. By the continuous embedding H s 0 (Ω) ֒→ L q (Ω), this yields u − n q → 0 as n → ∞. So we deduce that u n q is bounded in R. Using this fact, we want to show that (u n ) is bounded in H s 0 (Ω) as well. Since q < p < 2 * s in our assumptions, we can find τ ∈ (0, 1) s.t.
By the interpolation inequality (see [8, p. 93]) and the continuous embedding H s 0 (Ω) ֒→ L 2 * s (Ω), we have for all n ∈ N Since pτ < 2 we deduce that (u n ) is bounded in H s 0 (Ω). Now we conclude as in the proof of Theorem 3.3. Now we prove that ϕ + is unbounded from below. Indeed, letû 1 be defined as in Proposition 2.8 (i), and recall that û 1 2 = λ 1 s (Ω), û 1 2 2 = 1. By H 2 (iv), given θ > λ 1 s (Ω) 2 we can find K > 0 s.t. F (x, t) θt 2 for a.e. x ∈ Ω and all |t| > M . For all µ > 0 we have and the latter goes to −∞ as µ → ∞. So We claim that 0 is a local minimizer for ϕ + . By H 2 (ii) we have F + (x, t) 0 for a.e. x ∈ Ω and all |t| σ. For all u ∈ C 0 δ (Ω) with So, 0 is a local minimizer of ϕ + in C 0 δ (Ω). By Proposition 2.7, 0 is as well a local minimizer of ϕ + in H s 0 (Ω). As usual, it is not restrictive to assume that 0 is a strict local minimizer for both ϕ + and (reasoning as in the proof of (3.7)) there exists r > 0 s.t. ϕ + (γ(t)).
Using the critical groups, we can improve the conclusion of Proposition 4.2 under the same assumptions: Proof. Reasoning as in the proof of Proposition 4.2 we see that ϕ, ϕ ± satisfy (C), are unbounded from below and have a strict local minimum at 0. Moreover we know that 0, u ± ∈ K(ϕ). We aim at finding a further critical point for ϕ. We argue by contradiction, assuming In particular, all critical points of ϕ are isolated. Taking a < b in R s.t. all critical levels of ϕ lie in (a, b), from Theorem 2.12 we have Now we will compute all critical groups of ϕ both at its critical points and at infinity, then we will plug results into (4.6) to get a contradiction. In doing so, we will also need to compute some critical groups of ϕ ± . We begin with critical groups at infinity: for all integer k 0 we have We focus on ϕ (the argument for ϕ ± is analogous). We recall from the proof of Proposition 4.2 that We denote the unit sphere in H s 0 (Ω) by Reasoning as in the proof of (4.3) we see that for all u ∈ S Moreover, taking c < 0 small enough, we have for all v ∈ ϕ −1 (c) Indeed, by H 2 (v) there exists β, M > 0 s.t. f (x, t)t − 2F (x, t) β|t| q for a.e. x ∈ Ω and all |t| > M . Then, using also H 2 (i), for all v ∈ ϕ −1 (c) we have with a constant C M > 0 only depending on M . So, choosing we get (4.9). Now we apply the implicit function theorem [28,Theorem 7.3] to the function (µ, u) → ϕ(µu) defined in (1, ∞) × S. By (4.9) we have for all (µ, u) hence there exists a continuous mapping ρ : So we have ϕ c = {µu : u ∈ S, µ ∈ [ρ(u), ∞)}.
Reasoning as in the proof of Proposition 4.2 and using (4.5), we see that 0 is a strict local minimizer of ϕ, so (4.10) follows from (2.6) (the argument for ϕ ± is analogous). Finally we compute the critical groups at u ± : for all k 0 we have (4.11) C k (ϕ, u ± ) = δ k,1 R.
By (4.12), we get (4.11). Plugging (4.7), (4.10), and (4.11) into (4.6), we have ∞ k=0 (−1) k (δ k,0 + 2δ k,1 ) = 0, namely −1 = 0, a contradiction. Thus, (4.5) cannot hold, i.e., there exists a further critical point u ∈ K(ϕ) \ {0, u + , u − }. By Proposition 2.3, we see that u ∈ C 0 δ (Ω) and is a solution of (1.1). Remark 4.4. A comparison between Theorems 3.3 and 4.3 is now in order. Though formally the statements of such results coincide, the underlying structure of the critical set K(ϕ) is widely different in the two cases: in the sublinear case we have two local minimizers u + , u − and a third non-zero critical pointũ, typically of mountain pass type; while in the superlinear case we have two mountain pass-type points u + , u − and a third non-zero critical point of undetermined natureũ.
Aknowledgement. The second author is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).