Resonant problems for non-local elliptic operators with unbounded nonlinearites

: In this paper we study the existence of nontrivial solutions of a class of asymptotically resonant problems driven by a non-local integro-di ff erential operator with homogeneous Dirichlet boundary conditions by applying Morse theory and critical groups for a C 2 functional at both isolated critical points and infinity.


Introduction
In this paper we will study the existence of nontrivial solutions of the following nonlocal elliptic problem        −L K u = λ ℓ u + g(x, u) x ∈ Ω, where Ω is an open bounded subset of R N with a smooth boundary, g : Ω × R → R is a differential function whose properties will be given later, λ ℓ is an eigenvalue of L K and L K is a non-local elliptic operator formally defined as follows (1.5) The problem (1.5) can be regarded as the counterpart of the semilinear elliptic boundary value problem where λ ℓ is an eigenvalue of −∆ with a 0-Dirichlet boundary value. A weak solution for (1.1) is a function u : R N → R such that (1.7) Here the linear space X 0 = {v ∈ X : v = 0 a.e. in R N \ Ω}, and the functional space X denotes the linear space of Lebesgue measurable functions from R N to R such that the restriction to Ω of any function v in X belongs to L 2 (Ω) and the map (x, y) → (v(x) − v(y)) K(x − y) is in L 2 (R 2N \ (CΩ × CΩ), dxdy), where CΩ := R N \ Ω. The properties of the functional space X 0 will be introduced in the next section. The non-local 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 measured data, optimization, finance, etc. See [1] and the references therein. The non-local problems and operators have been widely studied in the literature and have attracted the attention of lot of mathematicians coming from different research areas due to the interesting analytical structure and broad applicability. Many mathematicians have applied variational methods [2] such as the mountain pass theorem [3], the saddle-point theorem [2] or other linking type of critical point theorem in the study of non-local equations with various nonlinearities that exhibit subcritical or critical growth; see [1,[4][5][6][7][8][9][10][11][12][13] and references therein.
In the present paper we will apply the Morse theory to find weak solutions to (1.1). We assume, throughout the whole paper, that the nonlinear function g ∈ C 1 (Ω × R, R) satisfies the following growth condition (g) there is C > 0 and p ∈ (2, 2N N−2s ) such that We consider the situation that the problem (1.1) has the trivial solution u ≡ 0 and is resonant at infinity in the sense that the function g satisfies the following assumptions We refer the reader to [ [11,Proposition 4] for the existence and basic properties of the eigenvalue of the linear non-local eigenvalue problem given by that will be collected in the next section.
We make some further conditions on g.
We will prove the following theorems. We first consider the case that g ′ t (x, 0)+λ ℓ is not an eigenvalue of (1.11). We have the following conclusions. Theorem 1.1. Assume (1.3), (1.9) and (g 1 ). Then the problem (1.1) admits at least one nontrivial weak solution in each of the following cases: For the case that g ′ t (x, 0) + λ ℓ = λ m , an eigenvalue of (1.11), i.e., the trivial solution u = 0 of (1.1), is degenerate. In this case the problem (1.1) is double resonant at both infinity and zero. We have the following conclusions.
Notice that in Theorem 1.2 there is a large difference between λ ℓ and λ m . This can be reduced by imposing on g some local sign conditions near zero. We denote f (x, t) := λ ℓ t + g(x, t) and F(x, t) = t 0 f (x, s)ds. We assume the following Then the problem (1.1) admits at least one nontrivial weak solution in each of the following cases: We give some remarks and comparisons. The non-local equations with resonance at infinity have been studied in some recent works. In [6,7], the famous saddle-point theorem [2] has been applied in the existence of solutions of the non-local problem related to (1.1) for the Landesman-Lazer resonance condition [15]. In [6], the authors treated a case in which one version of the Landesman-Lazer resonance condition [15] was formulated as follows: In [7], the authors treated an autonomous case in which another version of the Landesman-Lazer resonance condition [15] was formulated as follows: where E(λ ℓ ) is the linear space generated by the eigenfunctions corresponding to λ ℓ . In [7], there is a crucial assumption that all functions in E(λ ℓ ) having a nodal set with the zero Lebesgue measure, which is valid for the fractional Laplacian (−∆) s (see [16]) and is still open for the general non-local elliptic operator −L K (see [7,Equation (1.12)] and remarks therein).
We note here that the common feature in (1.12) and (1.13) is that the nonlinear term g is bounded. Motivated by previous works [6,7], we treat, in the present paper, the completely resonant case via the application of Morse theory and critical groups. The results in this paper are new in two aspects. On one hand, the nonlinear term g is indeed unbounded and by imposing on g the global conditions (g 1 ) and (g ± 2 ), we do not make the same assumption on the eigenfunctions of (1.11) as that in [7]. On the other hand, we explore a new application of the abstract results about critical groups at infinity that were built in [17] and modified in [18]. The conditions on g used here were first constructed in [19] for semilinear elliptic problems at resonance. Some of the above theorems may be regarded as the natural extension of local setting (1.6) to the non-local fractional setting.
We prove the main results via Morse theory [20,21] and critical group computations. Precisely, we will work under the abstract framework built in [17] and modified in [18]. In Section 2, we collect some preliminaries about the variational formulas related to (1.1). In Section 3, we give the proofs of the main theorems including some technical lemmas.

Preliminaries
In this section we will give the preliminaries for the variational structure of (1.1) and preliminary results in Morse theory.

The functional setting
We first recall some basic results on the functional X 0 mentioned in Section 1. The functional space X 0 is non-empty because C 2 0 (Ω) ⊂ X 0 (see [22,Lemma 11]), and it is endowed with the norm defined as (2.1) Furthermore, (X 0 , ∥ · ∥ X 0 ) is a Hilbert space with a scalar product (see [10,Lemmas 6 and 7]) defined by The norm (2.1) on X 0 is related to the so-called Gagliardo norm This embedding is compact whenever q ∈ [1, 2N N−2s ).

An eigenvalue problem for −L K
Next, we recall some basic facts about the eigenvalue problem associated with the integro- We denote by {λ k } k∈N the sequence of the eigenvalue of the problem (2.3), with 0 < λ 1 < λ 2 ⩽ · · · ⩽ λ k ⩽ · · · and λ k → +∞ as k → +∞.
We denote by ϕ k the eigenfunction corresponding to λ k . The sequence {ϕ k } k∈N can be normalized in such a way that the sequence provides an orthonormal basis of L 2 (Ω) and an orthogonal basis of X 0 . By [14,Proposition 2.4] one has that all ϕ k ∈ L ∞ (Ω). One can refer to [12, Proposition 9 and Appendix A], [14,Proposition 2.3] and [11,Proposition 4] for a complete study of the spectrum of the integrodifferential operator −L K . The first eigenvalue λ 1 is simple and can be characterized as Each eigenvalue λ k , k ⩾ 2, has finite multiplicity. More precisely, we say that λ k has the finite multi- The set of all of the eigenfunctions corresponding to λ k agrees with The eigenvalue λ 1 is achieved at a positive function ϕ 1 with ∥ϕ 1 ∥ L 2 (Ω) = 1. For each k ⩾ 2, the eigenvalue λ k can be characterized as follows: where P k := {u ∈ X 0 : ⟨u, ϕ j ⟩ X 0 = 0 for all j = 1, 2, · · · , k − 1}.
Corresponding to the eigenvalue λ k of −L K with multiplicity ν k , the space X 0 can be split as follows: For each eigenvalue λ k , we can define a linear operator A k : X 0 → X * 0 by By the continuous embedding from X 0 into L 2 (Ω) in Proposition 2.1, one can deduce that A k is a bounded self-adjoint linear operator so that ⟨A k ϕ, ϕ⟩ = 0 for all ϕ ∈ V k =: ker(A k ). Finally, we conclude this subsection with the following variational inequalities which can be deduced by the variational characterization of the eigenvalues and the standard Fourier decomposition:

Proofs of main results
In this section we give the proofs of the main results in this paper via some abstract results on Morse theory [20,21] for a C 2 functional J defined on a Hilbert space. These results come from [17,18,20,21,[23][24][25], etc. We refer the readers to [26] for a brief summary of the concepts, definitions and the abstract results about critical groups and Morse theory.
First of all, we observe that the problem (1.1) has a variational structure; indeed, it is the Euler-Lagrange equation of the functional J : X 0 → R defined as where G(x, t) = t 0 g(x, ς)dς. Since the nonlinear function g satisfies the assumption (g), by Proposition 2.1, the functional J is well defined on X 0 and is of class C 2 (see a detailed proof in [26]) with the derivatives given by According to (2.7) with λ ℓ , the functional J can be written as Using the assumption (g 1 ) we can deduce that Therefore J fits the basic assumptions in the abstract framework required by [26, Proposition 2.5] with respect to X 0 = V ℓ ⊕ W ℓ . Next we prove one technical lemma that will be used to verify the angle conditions required by [26, Proposition 2.5] for computation of the critical groups at infinity. . Then there exist M > 0, ϵ ∈ (0, 1) and β > 0 such that for any u = v + w ∈ X 0 = V ℓ ⊕ W ℓ with ∥u∥ X 0 ⩾ M and ∥w∥ X 0 ⩽ ϵ∥u∥ X 0 .
Proof. We give the proof for the case that (g 1 ) and (g + 2 ) hold.
For u ∈ C(M, ϵ), we have It follows from (g 1 ) and (g + 2 ) that (3.10) Since V ℓ is finite dimensional, by the elementary inequality |a + b| q ⩽ 2 q−1 (|a| q + |b| q ) for all a, b ∈ R, we have that hereĉ is the embedding constant of L 1+r (Ω) → V ℓ . Therefore for u ∈ C(M, ϵ), it follows from (3.9)- (3.12) Now we can take M > 0 large enough and 0 < ϵ < 1 small enough so that The proof is complete. □ In order to apply [26,Proposition 2.5] and Morse theory to prove our results, we have to verify that J satisfies the Palais-Smale condition. Proof. Let the sequence {u n } ⊂ X 0 be such that (3.14) We show that {u n } is bounded in X 0 . Suppose, by the way of contradiction, that Write u n = v n + w n , where v n ∈ V ℓ and w n ∈ W ℓ . By the variational inequalities (2.8) and (2.9), we have By (3.11), there is N 1 ∈ N such that |⟨J ′ (u n ), w n ⟩| ⩽ ∥w n ∥ X 0 , ∀ n ⩾ N 1 . (3.17)
□ Notice here that only (3.14) is used for verifying the Palais-Smale condition, it follows that the critical point set of J is compact and is then bounded. Now we are ready to give the proofs of the main results in this paper. Proof of Theorem 1.1. We give the proof of the case (i). Since it follows from Lemma 3.1 that J satisfies the angle condition (AC − ∞ ) in [26,Proposition 2.5] at infinity with respect to X 0 = V ℓ ⊕ W ℓ . Thus by [26,Proposition 2.5(ii)] we have C q (J, ∞) δ q,µ ℓ +ν ℓ Z, q ∈ Z, (3.30) where Therefore J has a critical point u * satisfying The second derivative of J at the trivial solution u = 0 can be written as By the condition we see that u = 0 is a nondegenerate critical point of J with the Morse index Since λ m λ ℓ , we get that µ ℓ + ν ℓ μ 0 , and we see from (3.33) and (3.34) that u * 0. The case (ii) can be proved in the same way. The proof is complete. □ Proof of Theorem 1.2 We give the proof of the case (ii). It follows from Lemma 3.1 that J satisfies the angle condition (AC + ∞ ) in [26,Proposition 2.5] at infinity with respect to X 0 = V ℓ ⊕ W ℓ . Thus by [26,Proposition 2.5(ii)] we have and J has a critical point u * satisfying Now J ′′ (0) takes the form It follows that 0 is a degenerate critical point of J with the Morse index µ 0 and the nullity ν 0 given by It follows from λ m < λ ℓ−1 < λ ℓ or λ ℓ−1 < λ m−1 that µ 0 + ν 0 < µ ℓ or µ 0 > µ ℓ , and we see from (3.36) and (3.39) that u * 0. The case (i) can be proved in the same way. The proof is complete. □ Lemma 3.3. Assume (1.3), (1.9), (g 1 ) and (F ± 0 ). Then (i) C q (J, 0) δ q,µ 0 +ν 0 Z for (F + 0 ) holds, (ii) C q (J, 0) δ q,µ 0 Z for (F − 0 ) holds, where µ 0 and ν 0 are given by (3.38).
Proof. We will apply [24, Proposition 2.3] to prove the results. We first note that by (g 1 ) and the last part in the proof of Lemma 3.2, the functional J verifies the bounded Palais-Smale condition which ensures the deformation property for computing C q (J, 0) (see [20,21]).
We treat the case (ii) for which (F − 0 ) holds. We will prove that J has the local linking structure at 0 as with respect to X 0 = E − ⊕ E + , where E − = W − m and E + = V m ⊕ W + m . We refer the readers to [28,29] for the concept of the local linking.
Therefore, J has a local linking structure with respect to E = E − ⊕ E + with µ 0 = dim E − . It follows from [24, Proposition 2.3] that C q (J, 0) δ q,µ 0 Z. The case (i) is proved in a similar and simpler way. The proof is complete. □ Proof of Theorem 1.3. We give the proof of the case (iv). As in the proof of Theorem 1.1(ii), we have gotten the following conclusion that J satisfies the angle condition (AC + ∞ ) in [26,Proposition 2.5] at infinity with respect to X 0 = V ℓ ⊕ W ℓ , and then that J has a critical point u * satisfying C µ ℓ (J, u * ) 0. (3.48) By (F − 0 ) and Lemma 3.3, J has a local linking at 0 with respect to X 0 = E − ⊕ E + . Thus it follows from [24, Proposition 2.3] that C q (J, 0) δ q,µ 0 Z. (3.49) By λ ℓ−1 < λ m−1 , we have that λ ℓ < µ 0 . It follows from (3.48) and (3.49) that u * 0. The other cases can be proved in the same way. The proof is complete. □ Remark 3.4. We conclude the paper with some remarks. 1) In Theorem 1.3, the result for one nontrivial solution is valid for f that is locally Lipschitz continuous with f ′ (x, 0) ≡ λ m being replaced by satisfying lim t→0 f (x,t) t = λ m . In this case, we have only J ∈ C 2−0 (X 0 , R) and no Morse index is involved. We have the critical groups at zero by applying the local linking theorem in [23] as follows: C µ 0 +ν 0 (J, 0) 0 for (F + 0 ) holds; C µ 0 (J, 0) 0 for (F − 0 ) holds. 2) In the case that λ ℓ = λ 1 and (g − 2 ) holds, we have that µ ℓ = 0 and C q (J, ∞) δ q,0 Z. It follows that C q (J, u * ) δ q,0 Z. (3.52) Indeed, (3.50) is equivalent to J being bounded from below and (3.51) is equivalent to u * being a local minimizer of J. Furthermore, in the case that C q (J, 0) 0 for some q ⩾ 1, we can apply [30, Theorem 2.1], i.e., the most general version of the three critical point theorem, to get two nontrivial solutions of (1.1).

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