Resonant problems for fractional Laplacian

In this paper we consider the following fractional Laplacian equation \begin{document} $ \left\{\begin{array}{ll} (-\Delta).s u=g(x, u) & x\in\Omega,\\ u=0, & x \in \mathbb{R}.N\setminus\Omega,\end{array} \right. $ \end{document} where $ s\in (0, 1)$ is fixed, $\Omega$ is an open bounded set of $\mathbb{R}.N$, $N > 2s$, with smooth boundary, $(-\Delta).s$ is the fractional Laplace operator. By Morse theory we obtain the existence of nontrivial weak solutions when the problem is resonant at both infinity and zero.

1. Introduction. The nonlocal equations have been experiencing impressive applications in different subjects, such as the thin obstacle problem, phase transitions, stratified materials, anomalous diffusion, crystal dislocation, soft thin films, semipermeable membranes and flame propagation, conservation laws, ultrarelativistic limits of quantum mechanics, quasigeostrophic flows, multiple scattering, minimal surfaces, materials science, water waves, elliptic problems with measure data, optimization, finance, etc. See [24] and the references therein. In the recent years many mathematicians have made efforts to apply the minimax methods ( [25]) such as the mountain pass theorem [1], the saddle point theorem [25] or other linking type of critical point theorems in the study of the non-local fractional Laplacian equations with different nonlinearities having subcritical or critical growth, see [3,4,7,8,14,15,16,23,24,26,27,29,30,32,33] and references therein.
In this paper we apply the Morse theory to study the non-local equations with the nonlinearity being asymptotically linear at both infinity and zero. Precisely, we deal with the problem where s ∈ (0, 1) is fixed, Ω is an open bounded subset of R N with Lipschitz boundary, N > 2s, and (−∆) s is the fractional Laplace operator, which (up to normalization factors) is defined as
(1.4) Where the functional space H s 0 (Ω) is defined (see [27,29]) as H s 0 (Ω) := {v ∈ H s (R N ) : v = 0 a.e. in R N \ Ω}, in which the functional space H s (R N ) denotes the fractional Sobolev space of the functions v ∈ L 2 (R N ) such that is in L 2 (R 2N , dxdy).
We will consider the problem (1.1) under the situation that the following asymptotically limits exist: We refer to Proposition 9 and Appendix A in [30] for the existence and basic properties of the eigenvalue of (1.7) that will be collected in the next section. We note here that (1.5) characterizes the problem (1.1) as asymptotically linear resonance at infinity while (1.6) characterizes the problem (1.1) is rasonant at zero, namely, the trivial solution of (1.1) is degenerate.
As we will deal with (1.1) at resonance, we need to make on the eigenvalues of (1.7) the following assumption: λ is an eigenvalue of (−∆) s such that all the eigenfunctions corresponding to λ have nodal set with zero Lebesgue measure. (1.8) It is well known that the assumption (1.8) is always valid for the classical Laplacian −∆ in a bounded domain Ω( [13,18]). For the fractional Laplacian (−∆) s , it makes sense at least when λ is the first eigenvalue of (−∆) s (see [30,31]). As ones have pointed out in [16] it is a quite interesting problem that whether or not (1.8) is valid for all the eigenvalues of (−∆) s . The problem (1.1) admits a trivial solution u = 0 due to g(x, 0) ≡ 0. The aim of this paper is to find nontrivial solutions of (1.1) under (1.5) and (1.6). The existence of nontrivial solutions for (1.1) depends on the interplay of the behaviors of g near zero and near infinity with the eigenvalues of (−∆) s , therefore further conditions are needed.
Set f (x, t) = g(x, t) − λ t and f 0 (x, t) := g(x, t) − λ m t. We make the following assumptions on f and f 0 . (f ± ) There are R > 1, r ∈ (0, 1) and c 1 , c 2 > 0 such that There are a large class of functions satisfying the above assumptions. We give an example as follows.
The main results of the present paper are stated as follows.
Theorem 1.2. Assume (g) and (1.8). Then the problem (1.1) admits at least one nontrivial weak solution in each of the following cases: and λ = λ m ; In particular, the conclusion is valid in (i) and (ii) for the case that λ = λ m . Theorem 1.3. Assume (g), (f − ) and (f + 0 ). If λ = λ m = λ 1 , then the problem (1.1) admits at least two nontrivial weak solutions. Now we give some remarks and comments. Resonance problems have received much attention in the literature since the appearance of a well-known work [19] by Landesman and Lazer. A pioneering condition is the famous Landesman-Lazer resonance condition introduced in [19] in studying semilinear elliptic problem via topological degree. A variational formulation for this condition was given by Rabinowitz (see [25]) in verifying the saddle point theorem.
The nonlocal fractional Laplacian problem with resonance may have its own meanings and has been considered in [14,15,16] by the saddle point theorem( [25]). In [14,16], some existence results related to (1.1) had been obtained for the Landesman-Lazer resonance setting( [19]). One version of the Landesman-Lazer resonance condition ( [19]) can be formulated as The main feature of (1.9) is the boundedness of nonlinear term f .
In the present paper we focus on applying Morse theory to (1.1) with resonance at both infinity and zero. The nonlinear term f is an unbounded function when it satisfies (f ± ). The type of conditions (f ± ) and (f ± 0 ) were first introduced in the works [21,20,34] for semilinear problems at resonance and was first used in [35] to find multiple solutions for the semilinear elliptic equation (1.10) The problem (1.10) may be seen as the one reduced to s = 1 from (1.1). In this sense Theorems 1.1 and 1.2 may be regarded as the fractional versions of the existence results in [35]. Theorem 1.3 is new. We prove the main results employing the Morse theory [12,22] and critical groups computations. Precisely, we will work under the abstract framework built in [6] and modified in [34,36]. The main difficulties in working such a framework are obviously related to the nonlocal nature of the problem.
The paper is organized as follows. In Section 2 we collect some preliminary in building the variational formulas related to (1.1) and then recall some abstract tools about Morse theory and critical groups. In Section 3 we give some technical lemmas for verifying conditions required by the abstract tools. In Section 4 the proofs of main results are given including further comments, comparisons and remarks.
2. Preliminary. In this section we will give the preliminaries for the variational structure of (1.1) and preliminary results in Morse theory.
2.1. Variational formulations for (1.1). We first recall some basic results on the functional spaces H s (R N ) and H s 0 (Ω). See [24,26,27,29,30,31,32] for details. The functional space H s (R N ) denotes the fractional Sobolev space of the functions u ∈ L 2 (R N ) such that The space H s (R N ) is endowed with the so-called Gagliardo norm Let H s 0 (Ω) be the function space defined as |u(x) − u(y)| 2 |x − y| N +2s dxdy 1 2 , and the scalar product By Lemma 8 in [27] and Lemma 9 in [32], we have following embedding results.
, the embedding H s 0 (Ω) → L q (R N ) is continuous and there is C q > 0 such that . This embedding is compact whenever q ∈ [1, 2N N −2s ).
We have the following variational formulations for (1.1) due to the above embedding.
Then I is well defined on H s 0 (Ω) and is belonging to C 2 (H s 0 (Ω), R) with derivatives given by Proof. We follow the arguments in [2]. From (g) we deduce that (2.2) By (2.2) and Proposition 2.1 we see that I is well defined on H s 0 (Ω). (i) We first check that I is Gâteaux differentiable on H s 0 (Ω). For each u ∈ H s 0 (Ω), we have to prove that for all v ∈ H s 0 (Ω), it holds that It is obvious that, for almost every x ∈ Ω, By the Lagrange Theorem there exists a real number θ such that |θ| | | and By the continuous embedding H s 0 (Ω) → L q (Ω) for all q ∈ [1, 2N N −2s ], we get easily that |v| + |u| p−1 |v| + |v| p ∈ L 1 (Ω), by the Dominated Convergence Theorem we have By (2.1) and Proposition 2.1, we deduce that Thus, as a function on v, the linear functional v → Ω g(x, u)vdx is a continuous on H s 0 (Ω), it is the Gâteaux differential of I at u on H s 0 (Ω), we denote it by I G , that is Now we check the continuity of the mapping I G : H s 0 (Ω) → (H s 0 (Ω)) * . To this aim, take a sequence {u n } in H s 0 (Ω) such that u n → u in H s 0 (Ω). By Proposition 2.1, up to a subsequence, we may assume that as n → ∞ and there exists ψ ∈ L p (R N ) such that We have, by the Hölder inequality, and moreover, here we have used the elementary inequality |a + b| q c q (|a| q + |b| q ) for all a, b ∈ R. Then by the Dominated Convergence Theorem, (2.7) By (2.5), (2.7) and Proposition 2.1 we get Therefore I G is continuous on H s 0 (Ω) and then we deduce that I is Fréchet differentiable on H s 0 (Ω) with derivative By the Lagrange Theorem there exists a real number θ such that |θ| | | and then it follows from the Dominated Convergence Theorem we have for all v, w ∈ H s 0 (Ω), defines a conjugate bilinear function on H s 0 (Ω), and by the Hölder inequality and the continuous embedding . Hence, by applying Riesz's representation theorem, I is Gâteaux differentiable at u with derivative Up to a subsequence, we may assume that a.e. in R n for any n ∈ N.
By the fact that the map t → g t (·, t) is continuous in t ∈ R we get lim n→∞ |g t (x, u n (x)) − g t (x, u(x))| = 0 a.e. in R N , moreover, by (1.3) and Proposition 2.1 we have we then obtain by the Dominated Convergence Theorem that and The proof is complete.
, and J can be written as it follows from Proposition 2.2 that under the assumption (g), the functional J ∈ C 2 (H s 0 (Ω), R) with derivatives given by (2.11) Therefore by (1.4) and (2.10), weak solutions of (1.1) are exactly critical points of the functional J on H s 0 (Ω).

2.2.
The eigenvalue problem (1.7). In this subsection we consider the eigenvalue problem mentioned in Section 1 and rewrite (1.7) as follows: The number λ ∈ R is an eigenvalue of (2.12) if there is a nontrivial function u ∈ H s 0 (Ω) such that From Proposition 9 in [30] we have the following conclusions. (i) (2.12) admits an eigenvalue λ 1 which is positive and that can be characterized as follows λ 1 is simple, and there is a non-negative function φ 1 ∈ H s 0 (Ω) corresponding to λ 1 , such that φ 1 L 2 (Ω) = 1 and (ii) the set of the eigenvalue of (2.12) consists of a sequence {λ k } k∈N with 0 < λ 1 < λ 2 · · · λ k · · · and λ k → ∞ as k → ∞.
For any k ∈ N the eigenvalues can be characterized as follows: and moreover, λ k is attained at some φ k ∈ P k , that is where for some ν k 1, then the set of all the eigenfunctions corresponding to λ k agrees with (iv) The sequence {φ k } k∈N of eigenfunctions corresponding to λ k is an orthonormal basis of L 2 (Ω) and an orthogonal basis of H s 0 (Ω).

YUTONG CHEN AND JIABAO SU
For each eigenvalue λ k , we define a linear operator A k : Then A k is a bounded self-adjoint linear operator, and corresponding to the eigenvalue λ k with multiplicity ν k , H s 0 (Ω) can be split as so that A k φ, φ = 0 for all φ ∈ V k := E(λ k ) and the following variational inequalities hold true: We conclude this subsection by citing an L ∞ -regularity result for the eigenfunctions of the fractional Laplacian (−∆) s .

2.3.
Preliminaries about Morse theory. In this subsection we collect some results on Morse theory(see [12,22]) for a C 2 functional J defined on a Hilbert space E.
We say that J possesses the deformation property at the level c ∈ R if for anȳ > 0 and any neighborhood N of K c , there are > 0 and a continuous deformation We say that J possesses the deformation property if J possesses the deformation property at each level c ∈ R. We say that J satisfies the Palais-Smale condition at the level c ∈ R if any sequence {u n } ⊂ E satisfying J (u n ) → c and J (u n ) → 0 as n → ∞ has a convergent subsequence. J satisfies the Palais-Smale condition if J satisfies the Palais-Smale condition at each c ∈ R. If J satisfies the Palais-Smale condition, then J possesses the deformation property( [12]).
Let u 0 be an isolated critical point of J with J (u 0 ) = c ∈ R, and U be a neighborhood of u 0 . The group C q (J , u 0 ) := H q (J c ∩ U, J c ∩ U \ {u 0 }), q ∈ Z, is called the q-th critical group of J at u 0 , where H * (A, B) denotes a singular relative homology group of the pair (A, B) with integer coefficients.
Let K = {u ∈ E : J (u) = 0}. Assume that J (K) is bounded from below by a ∈ R and J possesses the deformation property at all c a. The group C q (J , ∞) := H q (E, J a ), q ∈ Z, is called the q-th critical group of J at infinity( [6]).
Assume that J satisfies the deformation property and K is a finite set. The Morse type numbers of the pair (E, J a ) are defined by M q := u∈K dim C q (J , u), and the Betti numbers of the pair (E, J a ) are defined by β q := dim C q (J , ∞). If all M q and β q are finite and only finitely many of them are nonzero, then the relationship between the Morse type numbers M q and the Betti numbers β q is expressed by the following Morse inequalities ( [12,22] for some q ∈ Z then J must have a new critical point. Therefore the basic idea in applying Morse theory to find critical points of J is to compute critical groups both at infinity and at known critical points clearly and then to find unknown critical points by applying formulas (2.19) and (2.20). Now we state an abstract result for the critical groups at infinity.
Proposition 2.5(i) was obtained in [37](see Remark 5.2 in [12]) and Proposition 2.5(ii) is a revision of Proposition 3.10 in [6]. The revision was made first in [34] and was remade in [36].
Let the origin 0 be an isolated critical point of J ∈ C 2 (E, R) such that J (0) is a Fredholm operator, it is always the case in many applications. Let V 0 = ker J (0) and W 0 = V ⊥ 0 = W + 0 ⊕ W − 0 where W + 0 and W − 0 are positive definite and negative definite invariant subspaces of J (0), respectively. Then the nullity ν 0 := dim V 0 of 0 is finite(see [22]). If the Morse index µ 0 of J at 0 is finite, then we have the following basic facts(see [12,22]).
3. Some technical lemmas. In this section we prove some lemmas which will be used in the proof of main theorems. We first prove two simple facts related to the functions in H s 0 (Ω). These properties may be of their own meanings. Lemma 3.1. For any given σ > 0 small, there is a σ > 0 such that Proof. We argue by contradiction. Suppose that there exists σ 0 > 0 such that for each n ∈ N, there is u n ∈ H s 0 (Ω) such that meas{x ∈ Ω : |u n (x)| n u n H s 0 (Ω) } |Ω| − σ 0 . Then meas{x ∈ Ω : |u n (x)| n u n H s 0 (Ω) } σ 0 . Set Ω n = {x ∈ Ω : |u n (x)| n u n H s 0 (Ω) }. Then On the other hand, by Proposition 2.1 we have the continuous embedding H s 0 (Ω) → L 1 (Ω), it holds that Then we get C 1 nσ 0 . We get a contradiction by letting n → ∞.
The type of result similar to Lemma 3.1 were first given without proof in [34] and [35] where Hamiltonian systems and semilinear elliptic equations were treated. Here we provide a proof for the reader's convenient although the proof seems to be simple.

2)
Proof. We follow the arguments in [5]. Since E(λ) is the eigenfunction space of (−∆) s corresponding to the eigenvalue λ, we have by Proposition 2.3 that E(λ) is finite dimensional and by Proposition 2.4 that E(λ) ⊂ L ∞ (Ω).
For this, we argue once more by contradiction and we suppose that there were v 0 ∈ S such that for each n ∈ N, there exists w n ∈ E(λ) such that By (3.5), we get meas x ∈ Ω : |v 0 (x)| < 1 n τ, ∀ n ∈ N.
This contradicts the assumption (1.8), thus (3.4) is proved. We next consider the covering of Then for any v ∈ S ⊂ E(λ), it holds that v ∈ B bτ (vi) (v i ) for some i. Hence, as a consequence of this and (3.4), we get

This proves (3.3) and the proof is complete.
The type of result of Lemma 3.2 was a modification of a result in [5,Lemma 3.2]. From now on we prove two technical lemmas that will be used to verify the angle conditions required by Propositions 2.5 and 2.6 for computing the critical groups.
Let (1.5) hold. Then corresponding to the eigenvalue λ of (−∆) s with multiplicity ν , H s 0 (Ω) can be split as and . Then there exists M > 0, β > 0 and ∈ (0, 1) such that u H s 0 (Ω) . Proof. We give the proof for the case that (f + ) holds.
Using (f ± ) we can deduce that Indeed, by (f ± ) we deduce that Thus for u, φ ∈ H s 0 (Ω), As r ∈ (0, 1), (4.3) is verified. According to (2.13), the functional J can be written as Therefore J fits the the abstract framework required by Proposition 2.5. Now we verify that J satisfies the (PS) condition. Proof. Since H s 0 (Ω) is a reflexive Banach space and {u n } is bounded in H s 0 (Ω), there is a subsequence of {u n }, still denote by {u n }, and there exists u * ∈ H s 0 (Ω), such that u n u * weakly in H s 0 (Ω) as n → ∞.
(4.7) By Proposition 2.1, up to a subsequence, it holds that as n → ∞ and there exists ψ ∈ L p (R N ) such that |u n (x)| ψ(x) a.e. in R N for any n ∈ N, (4.9) where p ∈ (2, 2N N −2s ) was given in the assumption (g) which implies that g satisfies the growth condition (2.1).

That is lim
Finally we have that as n → ∞, thanks to (4.12) and (4.17). This proves (4.6) and the proof is complete.
. Then the functional J defined by (4.1) satisfies the Palais-Smale condition.
Proof. Let the sequence {u n } ⊂ H s 0 (Ω) be such that J (u n ) → 0, n → ∞. Write u n = v n + w n , where v n ∈ V ∞ and w n ∈ W ∞ . By (2.15) and (2.16), we have where By (4.18), there is N 1 ∈ N such that | J (u n ), w n | w n H s 0 (Ω) , ∀ n N 1 . It follows that for any given ε > 0 and all n sufficiently large, where M > 0 and ∈ (0, 1) was given in Lemma 3.3. Therefore by Lemma 3.3 we have that (4.27) On the other hand, by (4.18), we get (4.28) This contradicts (4.27). Hence {u n } is bounded in H s 0 (Ω). The proof is complete.
Now we are ready to give the proofs of Theorems 1.1-1.3. We denote Proof of Theorem 1.1. We give the proof for the case (i). By Lemma 4.2, the C 2 functional J satisfies the (PS) condition. At infinity, J takes the form (4.4) and fits the framework of Proposition 2.5. Since it follows from (f + ) and Lemma 3.3 that J satisfies the angle condition (AC − ∞ ) in Proposition 2.5 at infinity with respect to H s 0 (Ω) = V ∞ ⊕ W ∞ . Thus by Proposition 2.5(ii) we have C q (J , ∞) ∼ = δ q,µ +ν Z, q ∈ Z, (4.30) Therefore J has a critical point u * satisfying C µ +ν (J , u * ) ∼ = 0. (4.31) By (1.6) and (2.11) we see that Therefore by (2.13), near zero J takes the form where F 0 (x, t) = t 0 f 0 (x, ς)dς. Moreover, by (1.6) we see that 0 is a degenerate critical point of J with Morse index µ m and the nullity ν m . By Gromoll-Meyer result we have that C q (J , 0) ∼ = 0, for q ∈ [µ m , µ m + ν m ]. (4.34) By assumption, there is an eigenvalueλ of (1.7) such that λ <λ < λ m , it follows that µ + ν < µ m , we see from (4.31) and (4.34) that u * = 0. The proof is complete.
Proof of Theorem 1.3. In the case λ = λ m = λ 1 , we have that µ = µ m = 0 and ν = ν m = 1. With the same arguments, we deduce from (f − ) and Proposition 2.5(ii) that C q (J , ∞) ∼ = δ q,0 Z, q ∈ Z. It follows that ( [12]) C q (J , u * ) ∼ = δ q,0 Z. From (f + 0 ) and Proposition 2.6 we deduce that C q (J , 0) ∼ = δ q,1 Z, q ∈ Z. We conclude the paper with some comments, comparisons and remarks. This means that u * is a mountain pass point of J . In the classical semilinear Laplacian (1.10), from (4.43) one can deduce that C q (J , u * ) ∼ = δ q,1 Z. However, in the nonlocal fractional Laplacian setting, this is left open, to the best of our knowledge. Due to the nonlocal nature of the problem, many multiplicity results for (1.10) with resonance may not be easily extended to the nonlocal case. We will focus on the multiplicity results for the problem (1.1) in near future.
(2) The assumption (1.8) is necessary in this paper and can not be cancelled in the present setting. We note that (1.8) is always valid in the classical Laplacian (−∆) which may be regarded as deducing from (−∆) s to s = 1. The problem whether or not (1.8) is valid for each eigenvalue of (−∆) s is still open and is very interesting. One refers to [16] for some explanations.
(3) In this paper we have been working on the functional space H s 0 (Ω). We would like to point out that all the results and the arguments above are valid if one wants to work on a more general functional space in which the function |x| −(N +2s) is replaced by a function K : R N \ {0} → (0, ∞) with properties mK ∈ L 1 (R N ) with m(x) = min{|x| 2 , 1}, and there exists ϑ > 0 such that K(x) ϑ|x| −(N +2s) for any x ∈ R N \ {0}, as what have done in [7,8,14,15,16,23,26,27,29,30,32] and some references therein.