Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 85, pp. 1–21. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu MONOTONICITY PROPERTIES OF THE EIGENVALUES OF NONLOCAL FRACTIONAL OPERATORS AND THEIR APPLICATIONS GIOVANNI MOLICA BISCI, RAFFAELLA SERVADEI, BINLIN ZHANG Abstract. In this article we study an equation driven by the nonlocal inte- grodifferential operator −LK in presence of an asymmetric nonlinear term f . Among the main results of the paper we prove the existence of at least a weak solution for this problem, under suitable assumptions on the asymptotic be- havior of the nonlinearity f at ±∞. Moreover, we show the uniqueness of this solution, under additional requirements on f . We also give a non-existence re- sult for the problem under consideration. All these results were obtained using variational techniques and a monotonicity property of the eigenvalues of −LK with respect to suitable weights, that we prove along the present paper. This monotonicity property is of independent interest and represents the nonlocal counterpart of a famous result obtained by de Figueiredo and Gossez [14] in the setting of uniformly elliptic operators. 1. Introduction In recent years, a great attention has been focused on the study of fractional and nonlocal operators of elliptic type, both for the pure mathematical research and for concrete real-world applications. Fractional and nonlocal operators appear naturally in applications in many fields such as optimization, finance, phase transi- tions, stratified materials, anomalous diffusion, crystal dislocation, soft thin films, semipermeable membranes, flame propagation, conservation laws, ultra-relativistic limits of quantum mechanics, quasi-geostrophic flows, multiple scattering, minimal surfaces, materials science and water waves, see [3, 4, 6, 7, 8, 19, 31, 32, 33, 34] and the references therein. Also thanks to all these applications nonlocal fractional problems are widely studied in the literature. Many authors have considered nonlocal fractional Lapla- cian equations (and their generalizations) with different growth assumptions on the nonlinear term, such as superlinear and subcritical, critical, asymptotically linear and many others. We refer to the monograph [20] for an overview on these topics. There are a lot of interesting problems in the standard framework of the Lapla- cian and, more generally, of uniformly elliptic operators, widely studied in the literature. A natural question is whether or not the results got in this classical 2020 Mathematics Subject Classification. 35A01, 35S15, 47G20, 45G05. Key words and phrases. Fractional Laplacian; integrodifferential operator; nonlocal problems; eigenvalue and eigenfunction; asymmetric nonlinearities; variational methods; critical point theory; saddle point theorem. ©2022. This work is licensed under a CC BY 4.0 license. Submitted December 8, 2022. Published December 21, 2022. 1 2 G. MOLICA BISCI, R. SERVADEI, B. ZHANG EJDE-2022/85 context can be extended to the nonlocal framework of the fractional Laplacian type operators. In this spirit, in this article we are concerned with the existence, non-existence and uniqueness of solutions for the following nonlocal fractional equation with ho- mogeneous Dirichlet boundary conditions: −LKu = f(x, u) + g(x) in Ω u = 0 in Rn \ Ω . (1.1) Here Ω is an open bounded subset of Rn with Lipschitz boundary, n > 2s, s ∈ (0, 1), while LK is the integrodifferential operator defined as LKu(x) := ∫ Rn ( u(x+ y) + u(x− y)− 2u(x) ) K(y) dy , x ∈ Rn , (1.2) where the kernel K : Rn \ {0} → (0,+∞) is such that mK ∈ L1(Rn), with m(x) = min{|x|2, 1}; (1.3) there exists θ > 0 such that K(x) > θ|x|−(n+2s) for all x ∈ Rn \ {0} . (1.4) A typical model for K is the singular kernel K(x) = |x|−(n+2s) which gives rise to the fractional Laplace operator −(−∆)s, widely studied in the recent literature (see, for instance, the seminal papers [6, 32, 33]). Moreover, we suppose that f : Ω×R→ R is a Carathéodory function satisfying the following conditions there exist a1 ∈ L2(Ω) and a2 ∈ L∞(Ω) with a2 > 0 such that |f(x, t)| 6 a1(x) + a2(x)|t| for a.e. x ∈ Ω and for every t ∈ R ; (1.5) −∞ 6 α(x) := lim t→−∞ f(x, t) t and lim t→+∞ f(x, t) t =: β(x) 6 +∞ for a.e. x ∈ Ω . (1.6) Note that the asymptotic growth condition (1.6) on f at −∞ and +∞ includes also the case when the condition at −∞ is different from the one at +∞, in which case f is called asymmetric nonlinearity. Finally, the function g is such that g ∈ L2(Ω) . (1.7) In the classical context of the Laplacian and uniformly elliptic operators problems like (1.1) were widely studied in the literature, see, for instance, [1, 9]. In the context of nonlinear integral problems this kind of studies goes back to the classical results of Dolph [11]. Note that u ≡ 0 may not be a solution of (1.1), since f(x, 0)+g(x) may not van- ish. In this paper we prove the existence of weak solutions of (1.1) using variational methods. We denote by λ1 < λ2 6 . . . 6 λk 6 . . . the eigenvalues of the operator −LK in Ω with homogeneous Dirichlet boundary condition (see Subsection 2.2 for more details), and by Xs 0(Ω) the fractional func- tional space where we look for solutions (see Subsection 2.1 for a precise definition). The main results of this article can be stated as follows. EJDE-2022/85 MONOTONICITY PROPERTIES OF EIGENVALUES 3 Theorem 1.1. Let s ∈ (0, 1), n > 2s and Ω be an open bounded subset of Rn with Lipschitz boundary. Let K : Rn \ {0} → (0,+∞) satisfy assumptions (1.3) and (1.4) and let f : Ω×R→ R be a Carathéodory function satisfying (1.5) and (1.6), and g : Ω→ R be a function satisfying (1.7). Then, the following assertions hold: (i) if α(x), β(x) < λ1 for a.e. x ∈ Ω, then (1.1) has at least one weak solution in Xs 0(Ω). Moreover, the solution is unique if, in addition, for a.e. x ∈ Ω and for all t, t′ ∈ R, t 6= t′, 0 < f(x, t)− f(x, t′) t− t′ < λ1 (1.8) and the eigenfunctions of −LK corresponding to λ1 enjoy the unique con- tinuation property; (ii) if λk < α(x), β(x) < λk+1 for a.e. x ∈ Ω for some k ∈ N and the eigen- functions of −LK corresponding to λk and the ones corresponding to λk+1 enjoy the unique continuation property, then problem (1.1) has at least one weak solution in Xs 0(Ω). Moreover, the solution is unique if, in addition, for a.e. x ∈ Ω and for all t, t′ ∈ R, t 6= t′, λk < f(x, t)− f(x, t′) t− t′ < λk+1 ; (1.9) (iii) if either f(x, t) + g(x) < λ1t (1.10) or f(x, t) + g(x) > λ1t (1.11) for a.e. x ∈ Ω and for all t ∈ R, then (1.1) has no weak solution in Xs 0(Ω). Fiscella [16] obtained a similar result for the case α ≡ β and g ≡ 0. Therefore, [16, Theorem 1] can be viewed as a particular case of Theorem 1.1. In the setting of the fractional Laplacian problem (1.1) reads as follows (−∆)su = f(x, u) + g(x) in Ω u = 0 in Rn \ Ω , (1.12) where (−∆)s is the fractional Laplace operator defined, up to normalization factors, as − (−∆)su(x) = ∫ Rn u(x+ y) + u(x− y)− 2u(x) |y|n+2s dy x ∈ Rn . (1.13) In this article for k ∈ N we denote by λk,s the eigenvalues of (−∆)s in Ω with homogeneous Dirichlet boundary datum (see Subsection 2.2). In the framework of problem (1.12) we can state Theorem 1.1 as follows. Theorem 1.2. Let s ∈ (0, 1), n > 2s and Ω be an open bounded subset of Rn with Lipschitz boundary. Let f : Ω× R→ R be a Carathéodory function satisfying (1.5) and (1.6), and g : Ω → R be a function satisfying (1.7). Then the following assertions hold: (i) if α(x), β(x) < λ1,s for a.e. x ∈ Ω, then (1.12) has at least one weak solution in Hs(Rn). 4 G. MOLICA BISCI, R. SERVADEI, B. ZHANG EJDE-2022/85 Moreover, the solution is unique if, in addition, for a.e. x ∈ Ω and for all t, t′ ∈ R, t 6= t′, 0 < f(x, t)− f(x, t′) t− t′ < λ1,s ; (1.14) (ii) if λk,s < α(x), β(x) < λk+1,s for a.e. x ∈ Ω for some k ∈ N, then prob- lem (1.12) has at least one weak solution in Hs(Rn). Moreover, the solution is unique if, in addition, for a.e. x ∈ Ω and for all t, t′ ∈ R, t 6= t′, λk,s < f(x, t)− f(x, t′) t− t′ < λk+1,s ; (1.15) (iii) if either f(x, t) + g(x) < λ1,st (1.16) or f(x, t) + g(x) > λ1,st (1.17) for a.e. x ∈ Ω and for all t ∈ R, then (1.12) has no weak solution in Hs(Rn). In the classical setting it is well known that the interaction of α and β with the spectrum of the Laplace operator is closely related with the existence of weak solutions. Actually, Theorems 1.1 and 1.2 state that the absence of this interaction implies the existence of weak solutions for the fractional equations: this represents the nonlocal counterpart of the classical results by Dolph [11] (for other details we refer also to [1, 13]). However, there are a lot of cases where this interaction appears. In the seminal paper [2], Ambrosetti and Prodi firstly studied the case in which the derivative of nonlinearity jumps the first eigenvalue of the Laplacian operator (see also [1, 15, 23]). Hence a natural question arises: is there a weak solution in Xs 0(Ω) for problem (1.1) if α(x) < λ1 < β(x), or if λk < α(x) < λk+1 < β(x) a.e. in Ω? The answer is more delicate and still remains an open problem. This article is organized as follows. Section 2 is devoted to some preliminaries. In Section 3 we investigate an eigenvalues problem for −LK with weights, focusing on a comparison property of the eigenvalues which will be crucial in the proof of our main result. Here we also give a new result on the monotonicity property of the eigenvalues of the fractional Laplacian (−∆)s, which is of independent interest. In Section 4 we provide the proof of Theorem 1.1, using variational methods, together with the monotonicity result got in Subsection 3.1. Finally, in Section 5 we consider the case of the fractional Laplacian. 2. Preliminaries In this section we briefly introduce the notation and we recall some results for Sobolev fractional functional spaces used in this article. 2.1. Functional space Xs 0(Ω) and its properties. This subsection is devoted to the definition of the functional space Xs 0(Ω) introduced in [27] (see also [28, 29]) and we give some well known properties of it. The space Xs 0(Ω) is defined as Xs 0(Ω) := { g ∈ X : g = 0 a.e. in Rn \ Ω } , EJDE-2022/85 MONOTONICITY PROPERTIES OF EIGENVALUES 5 where X denotes the linear space of Lebesgue measurable functions from Rn to R such that the restriction to Ω of any function g in X belongs to L2(Ω) and the map (x, y) 7→ (g(x)− g(y)) √ K(x− y) is in L2 ( (Rn × Rn) \ (CΩ× CΩ) ) (here CΩ := Rn \ Ω). Note that the label s recalls that the kernel K satisfies (1.4) and that X and Xs 0(Ω) are non-empty, since C2 0 (Ω) ⊆ Xs 0(Ω), as proved in [27, Lemma 5.1]. We define a norm in Xs 0(Ω) as Xs 0(Ω) 3 g 7→ ‖g‖Xs 0 (Ω) := (∫ Rn×Rn |g(x)− g(y)|2K(x− y) dx dy )1/2 . (2.1) In this way, ( Xs 0(Ω), ‖ · ‖Xs 0 (Ω) ) is a Hilbert space (for this see [28, Lemma 7]), with scalar product 〈u, v〉Xs 0 (Ω) := ∫ Rn×Rn ( u(x)− u(y) )( v(x)− v(y) ) K(x− y) dx dy . (2.2) In the following we denote by Hs(Ω) the usual fractional Sobolev space endowed with the norm (the so-called Gagliardo norm) ‖g‖Hs(Ω) := ‖g‖L2(Ω) + (∫ Ω×Ω |g(x)− g(y)|2 |x− y|n+2s dx dy )1/2 . (2.3) We remark that, even in the model case in which K(x) = |x|−(n+2s), the norms in (2.1) and (2.3) are not the same, because Ω×Ω is strictly contained in Rn×Rn. This is the reason why the classical fractional Sobolev space approach is not sufficient for studying problem (1.1). In [30, Lemma 7] (see also [28, Lemma 5 and Lemma 6]) the authors proved the following result, which states a relation between the space Xs 0(Ω) and the usual fractional Sobolev spaces Hs(Rn). Lemma 2.1. The following assertions hold: (i) let K : Rn \ {0} → (0,+∞) satisfy assumptions (1.3) and (1.4). Then Xs 0(Ω) ⊆ Hs(Rn) and, moreover, for all v ∈ Xs 0(Ω) ‖v‖Hs(Ω) 6 ‖v‖Hs(Rn) 6 C‖v‖Xs 0 (Ω) , where C is a positive constant depending only on n, s, θ and Ω ; (ii) let K(x) = |x|−(n+2s). Then Xs 0(Ω) = { v ∈ Hs(Rn) : v = 0 a.e. in Rn \ Ω } . The following embedding result, proved in [28, Lemma 8] and in [30, Lemma 9], holds. Lemma 2.2. Let s ∈ (0, 1), n > 2s and Ω be an open bounded subset of Rn. Let K : Rn \ {0} → (0,+∞) satisfy (1.3) and (1.4). Then, the following assertions hold: (i) if Ω has a Lipschitz boundary, then the embedding Xs 0(Ω) ↪→ Lν(Rn) is compact for all ν ∈ [1, 2∗s), where 2∗s = 2n/(n− 2s) is the fractional critical Sobolev exponent; (ii) the embedding Xs 0(Ω) ↪→ L2∗s (Rn) is continuous. The counterpart of Lemma 2.2 in the usual fractional Sobolev spaces is given by the following one proved in [10, Theorem 6.5]. 6 G. MOLICA BISCI, R. SERVADEI, B. ZHANG EJDE-2022/85 Lemma 2.3. The embedding Hs(Rn) ↪→ Lν(Rn) is continuous for all ν ∈ [2, 2∗s]. 2.2. Eigenvalues and eigenfunctions of the operator −LK . This subsection deals with the following eigenvalue problem associated with the integrodifferential operator −LK : −LKu = λu in Ω u = 0 in Rn \ Ω . (2.4) We denote by {λk}k the sequence of the eigenvalues of (2.4), with 0 < λ1 < λ2 6 · · · 6 λk 6 . . . , λk → +∞ as k → +∞ (2.5) and by ek the eigenfunction corresponding to λk. When LK = −(−∆)s, the eigen- values and the eigenfunctions are denoted by λk,s and ek,s for all k ∈ N, respectively. Moreover, we normalize ek in such a way that the sequence {ek}k provides an orthonormal basis of L2(Ω) and an orthogonal basis of Xs 0(Ω). We know that λ1 is simple and e1 is non-negative (e1,s is positive as proved in [26, Corollary 8]). Finally, λ1 can be characterized as follows λ1 = min u∈X0\{0} ∫ Rn×Rn |u(x)− u(y)|2K(x− y)dx dy∫ Ω |u(x)|2 dx . (2.6) For a complete study of the spectrum of the integrodifferential operator −LK we refer to [24, Proposition 2.3], [29, Proposition 9 and Appendix A], and [25, Propo- sition 4]. Furthermore, we say that the eigenvalue λk, k > 2, has multiplicity m ∈ N if λk−1 < λk = · · · = λk+m−1 < λk+m . Then, the set of all the eigenvalues corresponding to λk agrees with span{ek, . . . , ek+m−1} . Finally, for all k ∈ N in the sequel we denote the spaces Hk := span{e1, . . . , ek}, (2.7) Pk+1 := {u ∈ Xs 0(Ω) : 〈u, ej〉Xs 0 (Ω) = 0 for j = 1, . . . , k}, (2.8) where 〈·, ·〉Xs 0 (Ω) is defined by (2.2). When LK = −(−∆)s, the spaces Hk and Pk+1 are denoted by Hk,s and Pk+1,s for all k ∈ N, respectively. Using definitions (2.7) and (2.8), the variational characterization of the eigen- values of −LK (see [29, Proposition 9] and [24, Proposition 2.3]) implies that∫ Rn×Rn |u(x)− u(y)|2K(x− y) dx dy > λk+1 ∫ Ω |u(x)|2 dx for all u ∈ Pk+1 (2.9) and∫ Rn×Rn |u(x)− u(y)|2K(x− y) dx dy 6 λk ∫ Ω |u(x)|2 dx for all u ∈ Hk (2.10) for all k ∈ N. EJDE-2022/85 MONOTONICITY PROPERTIES OF EIGENVALUES 7 3. An eigenvalue problem for −LK with weights In this section we are concerned with an eigenvalue problem for −LK with weights, whose properties will be used all along the paper. Precisely, we are inter- ested in the eigenvalue problem −LKu = λr(x)u in Ω u = 0 in Rn \ Ω , (3.1) where r : Ω→ R is such that r ∈ Lip(Ω), (3.2) r > 0 in Ω. (3.3) The weak formulation of problem (3.1) is∫ Rn×Rn (u(x)− u(y))(ϕ(x)− ϕ(y))K(x− y) dx dy = λ ∫ Ω r(x)u(x)ϕ(x) dx ∀ϕ ∈ Xs 0(Ω) u ∈ Xs 0(Ω). (3.4) We say that λ[r] ∈ R is an eigenvalue of −LK with weight r if there exists a non-trivial solution u ∈ Xs 0(Ω) of (3.1) with λ = λ[r]. In this case, u is called eigenfunction corresponding to the eigenvalue λ[r]. The existence of a sequence of eigenvalues λ1[r] < λ2[r] 6 . . . 6 λk[r] 6 . . . and of the corresponding eigenfunctions ek[r] of (3.1) was proved in [17, Proposi- tion 2.1] (see also [29, Proposition 9 and Appendix A]), under the assumption that the weight r satisfies (3.2) and (3.3). We refer to [17, Proposition 2.1] also for the following variational characterization of the eigenvalues and properties of the related eigenfunctions. The first eigenvalue λ1[r] of problem (3.1) is simple and it is given by λ1[r] = min u∈Xs 0 (Ω)\{0} ∫ Rn×Rn |u(x)− u(y)|2K(x− y) dx dy∫ Ω r(x)|u(x)|2 dx (3.5) and there exists a non-negative function e1[r] ∈ Xs 0(Ω), which is an eigenfunction corresponding to λ1[r], attaining the minimum in (3.5), that is∫ Ω r(x)|e1[r](x)|2dx = 1 λ1[r] = ∫ Rn×Rn |e1[r](x)− e1[r](y)|2K(x− y)dx dy . (3.6) Furthermore, for all k ∈ N the eigenvalues of (3.1) can be characterized as follows: λk+1[r] = min u∈Pk+1[r]\{0} ∫ Rn×Rn |u(x)− u(y)|2K(x− y)dx dy∫ Ω r(x) |u(x)|2 dx , (3.7) where Pk+1[r] := { u ∈ Xs 0(Ω) : 〈u, ej [r]〉Xs 0 (Ω) = 0, ∀j = 1, . . . , k } , 8 G. MOLICA BISCI, R. SERVADEI, B. ZHANG EJDE-2022/85 and by λk[r] = max u∈Hk[r]\{0} ∫ Rn×Rn |u(x)− u(y)|2K(x− y)dx dy∫ Ω r(x) |u(x)|2 dx , (3.8) where Hk[r] := span { e1[r], . . . , ek[r] } . Finally, we say that the eigenvalue λk[r], k > 2, has multiplicity m ∈ N if λk−1[r] < λk[r] = . . . = λk+m−1[r] < λk+m[r] , and, in this case, the set of all the eigenfunctions corresponding to λk[r] agrees with span{ek[r], . . . , ek+m−1[r]}. In [18, Proposition 2.1] the authors obtained another interesting characterization of the eigenvalues λk[r], which is the extension to the nonlocal fractional setting of [14, formula (3)] valid for uniformly elliptic operators. 3.1. Monotonicity properties of the eigenvalues of nonlocal operators with respect to the weights. This subsection is devoted to another important property of the eigenvalues of nonlocal fractional operators. In particular we deal with a monotonicity property of eigenvalues of problem (3.1) with respect to the weights and we obtain a result of independent interest. Firstly we recall that, as a consequence of [18, Proposition 2.1], Frassu and Iannizzotto proved a property of monotone dependence of the eigenvalues λk[r] with respect to the weights, provided the eigenvalues ek[r] satisfy a unique continuation property. We say that a family of functions has the unique continuation property if no function, besides possibly the zero function, vanishes on a set of positive Lebesgue measure. With this definition, we can state the following monotonicity property (with respect to the weights) for the eigenvalues, proved in [18, Theorem 3.2]. Proposition 3.1. Let s ∈ (0, 1), n > 2s and Ω be an open bounded subset of Rn with Lipschitz boundary and let K : Rn \ {0} → (0,+∞) satisfy assumptions (1.3) and (1.4). Let r1, r2 ∈ L∞(Ω) be such that 0 6≡ r1 6 r2 a.e. in Ω and r1 6≡ r2. Assume that either the eigenfunctions of (3.1) corresponding to λk[r1] or the ones corresponding to λk[r2] enjoy the unique continuation property for some k ∈ N. Then, λk[r1] > λk[r2]. Now, let us consider the eigenvalue problem (3.1) in the case K(x) = |x|−(n+2s), that is the following one (−∆)su = λr(x)u in Ω u = 0 in Rn \ Ω . (3.9) The main result of this subsection can be stated as follows. Proposition 3.2. Let s ∈ (0, 1), n > 2s and let Ω be a bounded domain of Rn with Lipschitz boundary. Let r1, r2 ∈ C1(Ω) ∩L∞(Ω) be such that 0 6≡ r1 6 r2 in Ω and r1 6≡ r2, and let λk,s[ri] be the eigenvalue of (3.9) with r = ri, i = 1, 2, for some k ∈ N. Then λk,s[r1] > λk,s[r2]. Proof. By [12, Theorem 1.4] and the regularity assumptions on r1 and r2 we know that the eigenfunctions ek,s[r1] and ek,s[r2] satisfy the unique continuation property. Hence, by Proposition 3.1 we obtain the assertion of Proposition 3.2. � EJDE-2022/85 MONOTONICITY PROPERTIES OF EIGENVALUES 9 Proposition 3.2 can be seen as the nonlocal counterpart of the well known result due to de Figueiredo and Gossez [14, Proposition 1]. Note that Proposition 3.2 improves [18, Corollary 4.2], where the authors consider just the case when s ∈ [1/4, 1). It is an open question if Proposition 3.2 holds for a general nonlocal fractional operator. As far as we know there are no result in the literature on the unique continuation property for the eigenfunctions of problem (3.1) 4. Variational formulation of the problem and beyond This section is devoted to problem (1.1). In Theorem 1.1 we state existence, uniqueness and non-existence results for it. These results are obtained by using variational methods, together with the monotonicity property given for the eigen- values of −LK in Proposition 3.1. First of all, we recall that, thanks to [27, Lemma 5.6] (see also [30, footnote 3]), by a weak solutions of (1.1) we mean a solution of the problem∫ Rn×Rn (u(x)− u(y))(ϕ(x)− ϕ(y))K(x− y)dx dy = ∫ Ω f(x, u(x))ϕ(x)dx+ ∫ Ω g(x)ϕ(x)dx ∀ϕ ∈ Xs 0(Ω) u ∈ Xs 0(Ω) . (4.1) We observe that problem (4.1) has a variational structure, indeed it is the Euler- Lagrange equation of the functional J : Xs 0(Ω)→ R defined as follows J (u) = 1 2 ∫ Rn×Rn |u(x)− u(y)|2K(x− y) dx dy− ∫ Ω F (x, u(x))dx− ∫ Ω g(x)u(x)dx, where F (x, t) = ∫ t 0 f(x, τ) dτ . (4.2) Note that the functional J is well-defined in Xs 0(Ω), because of assumptions (1.5) on the nonlinear term f , (1.7) on g, and Lemma 2.2. Furthermore, it is well known that the functional J is Frechét differentiable in Xs 0(Ω) and for all ϕ ∈ Xs 0(Ω), 〈J ′(u), ϕ〉 = ∫ Rn×Rn (u(x)− u(y))(ϕ(x)− ϕ(y))K(x− y) dx dy − ∫ Ω f(x, u(x))ϕ(x)dx− ∫ Ω g(x)ϕ(x)dx. Thus, critical points of the functional J are solutions of (4.1). Hence, our goal consists in looking for critical points of J . For this purpose we use Weierstrass Theorem in the case in which the functions α and β given in (1.6) are less than the first eigenvalue of problem (2.4), while we perform the Saddle Point Theorem by Rabinowitz (see [21, 22]) when α and β lie between two consecutive eigenvalues of −LK . Before going on, we observe that, as a consequence of (1.5), it is easy to prove that |F (x, t)| 6 a1(x)|t|+ a2(x) 2 |t|2 for a.e. x ∈ Ω and for all t ∈ R . (4.3) 10 G. MOLICA BISCI, R. SERVADEI, B. ZHANG EJDE-2022/85 Moreover, by (1.6) we obtain that −∞ 6 α(x) 2 = lim t→−∞ F (x, t) t2 and lim t→+∞ F (x, t) t2 = β(x) 2 6 +∞ (4.4) for a.e. x ∈ Ω. Now we are ready to prove Theorem 1.1. 4.1. Proof of Theorem 1.1-(i). In this subsection we consider the case when α(x), β(x) < λ1 for a.e. x ∈ Ω , (4.5) where λ1 is defined as in (2.6). Note that in this setting β < +∞, while α may possibly take the value −∞. Proof of Theorem 1.1-(i). Our strategy consists in applying the Weierstrass Theo- rem to the functional J . We proceed by steps. Step 4.1. The functional J is weakly lower semicontinuous in Xs 0(Ω). For this purpose we claim that the maps u 7→ ∫ Ω F (x, u(x))dx and u 7→ ∫ Ω g(x)u(x) dx (4.6) are continuous in the weak topology of Xs 0(Ω). For this, let {uj}j be a sequence in Xs 0(Ω) such that uj → u∞ weakly in Xs 0(Ω) as j → +∞. Then, by Lemma 2.2, up to a subsequence, still denoted by uj , we have uj → u∞ in Lν(Rn) for all ν ∈ [1, 2∗s), (4.7) uj → u∞ a.e. in Rn (4.8) as j → +∞ and, for all ν ∈ [1, 2∗s), there exists qν ∈ Lν(Rn) such that |uj(x)| 6 qν(x) for a.e. in Rn (4.9) see, for instance [5, Theorem IV.9]. Hence, by (1.7), (4.3), (4.7), (4.8), and the Dominated Convergence Theorem we deduce (4.6). Taking into account that the function u 7→ ‖u‖Xs 0 (Ω) is weakly lower semicontin- uous in Xs 0(Ω), by (4.6) we obtain that J is weakly lower semicontinuous in Xs 0(Ω) and this proves Step 4.1. Step 4.2. The functional J is coercive in Xs 0(Ω). Let {uj}j be a sequence in Xs 0(Ω) such that ‖uj‖Xs 0 (Ω) → +∞ (4.10) as j → +∞. Thus, up to a subsequence, there exists v∞ ∈ Xs 0(Ω) such that vj := uj ‖uj‖Xs 0 (Ω) → v∞ weakly in Xs 0(Ω) (4.11) as j → +∞ and, by the weak lower semicontinuity of the norm, ‖v∞‖Xs 0 (Ω) 6 1 . (4.12) By (4.11) and Lemma 2.2 it easily follows that vj → v∞ weakly in Xs 0(Ω) vj → v∞ in Lν(Rn) for all ν ∈ [1, 2∗s) vj → v∞ a.e. in Rn (4.13) EJDE-2022/85 MONOTONICITY PROPERTIES OF EIGENVALUES 11 as j → +∞ and, for all ν ∈ [1, 2∗s), there exists qν ∈ Lν(Ω) such that |uj(x)| ‖uj‖Xs 0 (Ω) 6 qν(x) for a.e. x ∈ Rn (4.14) for all j ∈ N (see, for example [5, Theorem IV.9]). By (4.3) we obtain that 0 6 |F (x, uj(x))| ‖uj‖2Xs 0 (Ω) 6 a1(x) |uj(x)| ‖uj‖2Xs 0 (Ω) + a2(x) 2 |uj(x)|2 ‖uj‖2Xs 0 (Ω) , which implies that lim j→+∞ F (x, uj(x)) ‖uj‖2Xs 0 (Ω) = 0 for a.e. x ∈ Ω provided that v∞(x) = 0 , (4.15) thanks to (4.10) and (4.13). Now, suppose that v∞(x) 6= 0. Then, again by (4.10) and (4.13) we obtain that |uj(x)| = |uj(x)| ‖uj‖Xs 0 (Ω) ‖uj‖Xs 0 (Ω) → +∞ (4.16) as j → +∞. Hence, by (4.4), (4.13), and (4.16), we have F (x, uj(x)) ‖uj‖2Xs 0 (Ω) = F (x, uj(x)) |uj(x)|2 |uj(x)|2 ‖uj‖2Xs 0 (Ω) → r(x) 2 |v∞(x)|2 (4.17) as j → +∞, where r(x) := { β(x) if v∞(x) > 0 α(x) if v∞(x) < 0. (4.18) By (4.17) and the fact that r(x) < λ1 for a.e. x ∈ Ω (see (4.5)), we deduce that lim j→+∞ F (x, uj(x)) ‖uj‖2Xs 0 (Ω) < λ1 2 |v∞(x)|2 for a.e. x ∈ Ω provided that v∞(x) 6= 0 . (4.19) All in all (4.15) and (4.19) give lim j→+∞ F (x, uj(x)) ‖uj‖2Xs 0 (Ω) 6 λ1 2 |v∞(x)|2 (4.20) for a.e. x ∈ Ω, with strict inequality if v∞(x) 6= 0. Finally, by (2.6), (4.3), (4.10), (4.13), (4.14), (4.20), and the Dominated Conver- gence Theorem, we obtain L := lim j→+∞ J (uj) ‖uj‖2Xs 0 (Ω) = lim j→+∞ (1 2 − ∫ Ω F (x, uj(x)) ‖uj‖2Xs 0 (Ω) dx− ∫ Ω g(x)uj(x) ‖uj‖2Xs 0 (Ω) dx ) > 1 2 − λ1 2 ∫ Ω |v∞(x)|2 dx (with strict inequality if v∞ 6≡ 0) > 1 2 − 1 2 ‖v∞‖2Xs 0 (Ω) > 0 (with strict inequality if v∞ ≡ 0) , (4.21) 12 G. MOLICA BISCI, R. SERVADEI, B. ZHANG EJDE-2022/85 since (4.12) holds. Thus L > 0 and by (4.21) we obtain that J (uj) > L 2 ‖uj‖2Xs 0 (Ω) for j sufficiently large. This, together with (4.10), yields that the functional J is coercive. This completes the proof of Step 4.2. By Steps 4.1 and 4.2, it is easy to see that J has a minimum u ∈ Xs 0(Ω), thanks to Weierstrass Theorem. Of course u is a weak solution of problem (1.1). Now, it remains to prove that u is unique, provided (1.8) is satisfied. For this, let u1, u2 ∈ Xs 0(Ω) be two distinct weak solutions of problem (1.1) and let v = u1−u2. Then, v 6≡ 0 in Ω and∫ Rn×Rn (v(x)− v(y))(ϕ(x)− ϕ(y))K(x− y)dx dy = ∫ Ω ( f(x, u1(x))− f(x, u2(x)) ) ϕ(x)dx = ∫ Ω r(x) ( u1(x)− u2(x) ) ϕ(x)dx = ∫ Ω r(x)v(x)ϕ(x)dx (4.22) for all ϕ ∈ Xs 0(Ω), where r(x) := { f(x,u1(x))−f(x,u2(x)) u1(x)−u2(x) if u1(x) 6= u2(x) 0 if u1(x) = u2(x). Note that r is a measurable function non identically zero and, by (1.8), 0 6 r(x) < λ1 for a.e. x ∈ Ω . (4.23) Hence, r ∈ L∞(Ω), being Ω bounded. Since v 6≡ 0 in Ω, by (4.22) we deduce that v is an eigenfunction of problem (3.1) whose corresponding eigenvalue is 1. Thus, there exists k̃ ∈ N such that v = ek̃[r] and the corresponding eigenvalue λk̃[r] is such that λk̃[r] = 1. Proposition 3.1, the fact that the eigenfunction of −LK corresponding to λ1 enjoys the unique continuation property (by assumption), and (4.23) yield that 1 = λk̃[r] > λk̃[λ1] = λk̃ λ1 , (4.24) thanks to the definition of λk̃[λ1]. By (4.24) we obtain the contradiction λ1 > λk̃ and so v ≡ 0 in Ω and the proof of Theorem 1.1-(i) is complete. � 4.2. Proof of Theorem 1.1-(ii). In this subsection we focus on the situation when α and β in (1.6) satisfy the condition λk < α(x), β(x) < λk+1 for a.e. x ∈ Ω and some k ∈ N , (4.25) where λj is the j-th eigenvalue of problem (2.4), j ∈ N. Proof of Theorem 1.1-(ii). First of all, let us prove the existence of at least one weak solution for problem (1.1): our aim in this setting is to perform the Saddle Point Theorem (see [22]). For this purpose, we proceed by steps. Step 4.3. All the Palais-Smale sequences for J are bounded in Xs 0(Ω). EJDE-2022/85 MONOTONICITY PROPERTIES OF EIGENVALUES 13 Let {uj}j be a Palais-Smale sequence for J . Assume, by contradiction, that ‖uj‖Xs 0 (Ω) → +∞ (4.26) as j → +∞. We have |〈J ′(uj), ϕ〉| 6 εj‖ϕ‖Xs 0 (Ω) for all ϕ ∈ Xs 0(Ω), (4.27) where εj → 0 as j → +∞. In particular, setting vj := uj/‖uj‖Xs 0 (Ω), by (4.26) and (4.27) we have lim j→+∞ ∫ Rn×Rn ( (vj(x)− vj(y) )( ϕ(x)− ϕ(y) ) K(x− y) dx dy − ∫ Ω f(x, uj(x)) ‖uj‖Xs 0 (Ω) ϕ(x)dx− ∫ Ω g(x) ‖uj‖Xs 0 (Ω) ϕ(x)dx ) = 0. (4.28) From the fact that {vj}j is bounded in Xs 0(Ω) and Lemma 2.2, up to a subsequence, still denoted by vj , we can assume that there exists v∞ ∈ Xs 0(Ω) such that vj → v∞ weakly in Xs 0(Ω) vj → v∞ in L2(Rn) vj → v∞ a.e. in Rn (4.29) as j → +∞, and there exists q ∈ L2(Ω) such that∣∣vj(x) ∣∣ 6 q(x) for a.e. x ∈ Rn (4.30) for all j ∈ N, see, for instance [5, Theorem IV.9]. Taking into account (1.5) we have 0 6 |f(x, uj(x))| ‖uj‖Xs 0 (Ω) 6 a1(x) ‖uj‖Xs 0 (Ω) + a2(x)|uj(x)| ‖uj‖Xs 0 (Ω) , which implies that lim j→+∞ f(x, uj(x)) ‖uj‖Xs 0 (Ω) = 0 for a.e. x ∈ Ω provided that v∞(x) = 0 , (4.31) thanks to (4.26) and (4.29). Now, let us consider the case when v∞(x) 6= 0. By (1.6) there exists a function h such that f(x, t) = β(x)t+ + α(x)t− + h(x, t), (4.32) with lim |t|→+∞ h(x, t) t = 0 , (4.33) where t+ := max{t, 0} and t− := min{t, 0}. Moreover, by (4.26) we obtain that |uj(x)| → +∞ a.e. x ∈ Ω as j → +∞ . (4.34) Therefore, by (4.29), (4.32), (4.33) and (4.34), we deduce that f(x, uj(x)) ‖uj‖Xs 0 (Ω) = β(x)v+ j (x) + α(x)v−j (x) + h(x, uj(x)) ‖uj‖Xs 0 (Ω) = β(x)v+ j (x) + α(x)v−j (x) + h(x, uj(x)) |uj(x)| |uj(x)| ‖uj‖Xs 0 (Ω) → β(x)v+ ∞(x) + α(x)v−∞(x) (4.35) a.e. x ∈ Ω, provided that v∞(x) 6= 0. 14 G. MOLICA BISCI, R. SERVADEI, B. ZHANG EJDE-2022/85 All in all, by (4.31) and (4.35), we have f(x, uj(x)) ‖uj‖Xs 0 (Ω) → β(x)v+ ∞(x) + α(x)v−∞(x) (4.36) a.e. x ∈ Ω. Then, by (4.36), the Dominated Convergence Theorem and again by (4.29) and (4.30), we obtain that lim j→+∞ ∫ Ω f(x, uj(x)) ‖uj‖Xs 0 (Ω) ϕ(x) dx = ∫ Ω ( β(x)v+ ∞(x) + α(x)v−∞(x) ) ϕ(x) dx (4.37) for all ϕ ∈ Xs 0(Ω). Taking into account (4.26), (4.28), and (4.37), we obtain∫ Rn×Rn ( v∞(x)− v∞(y) )( ϕ(x)− ϕ(y) ) K(x− y) dx dy = ∫ Ω ( β(x)v+ ∞(x) + α(x)v−∞(x) ) ϕ(x)dx for all ϕ ∈ Xs 0(Ω). This means that the function v∞ satisfies weakly (3.1) with λ = 1 and the weight r given by r(x) := { β(x) if v∞(x) > 0 α(x) if v∞(x) < 0. (4.38) Note that, since (4.25) holds, it follows that λk < r(x) < λk+1 for a.e. x ∈ Ω (4.39) and so r ∈ L∞(Ω), since Ω is bounded. Now we claim that v∞ ≡ 0 in Ω . (4.40) To prove this we argue by contradiction and we suppose that v∞ 6≡ 0. Then v∞ is an eigenfunction of (3.1) whose corresponding eigenvalue is 1, that is there exists k̃ ∈ N such that v∞ = ek̃[r] and the corresponding eigenvalue λk̃[r] = 1. Since, by assumption, the eigenfunctions of −LK corresponding to λk and the ones corresponding to λk+1 enjoy the unique continuation property, by Proposi- tion 3.1 and (4.39), we deduce that λk̃ λk+1 = λk̃[λk+1] < λk̃[r] = 1 < λk̃[λk] = λk̃ λk , (4.41) taking into account the definition of λk̃[λj ], j ∈ N. Hence, (4.41) yields that λk̃ ∈ (λk, λk+1). This is a contradiction, since (λk, λk+1) does not contain any eigenvalue of −LK . Thus, (4.40) holds and our claim is proved. By (4.28) with ϕ = vj , (4.29), (4.37), and (4.40), we deduce that 0 = lim j→+∞ 〈J ′(uj), uj〉 ‖uj‖2Xs 0 (Ω) = 1− lim j→+∞ ∫ Ω f(x, uj(x)) ‖uj‖2Xs 0 (Ω) uj(x)dx− lim j→+∞ ∫ Ω g(x) ‖uj‖2Xs 0 (Ω) uj(x)dx = 1 which is absurd. Therefore, the sequence {uj}j is bounded in Xs 0(Ω) and this proves Step 4.3. EJDE-2022/85 MONOTONICITY PROPERTIES OF EIGENVALUES 15 Step 4.4. All Palais-Smale sequences for J have a convergent subsequence in Xs 0(Ω). The proof is quite standard. We repeat it just to make this article selfcontained. Let {uj}j be a Palais-Smale sequence for J in Xs 0(Ω). By Step 4.3, the sequence {uj}j is bounded in Xs 0(Ω). Hence, since Xs 0(Ω) is a Hilbert space, up to a subse- quence, still denoted by uj , there exists u∞ ∈ Xs 0(Ω) such that uj → u∞ weakly in Xs 0(Ω) uj → u∞ in Lν(Rn) for all ν ∈ [1, 2∗s) uj → u∞ a.e. in Rn (4.42) as j → +∞ and, for all ν ∈ [1, 2∗s), there exists qν ∈ Lν(Ω) such that∣∣uj(x) ∣∣ 6 qν(x) a.e. x ∈ Rn (4.43) for all j ∈ N. By (1.5), (4.42), (4.43), and the Lebesgue Dominated Convergence Theorem, we obtain that ∫ Ω f(x, uj(x))uj(x)dx→ ∫ Ω f(x, u∞(x))u∞(x) dx∫ Ω f(x, uj(x))u∞(x) dx→ ∫ Ω f(x, u∞(x))u∞(x) dx (4.44) as j → +∞, while, by (1.7) and (4.42) we obtain∫ Ω g(x)uj(x) dx→ ∫ Ω g(x)u∞(x) dx (4.45) as j → +∞. Moreover, since {uj}j is a Palais-Smale sequence, we know that 〈J ′(uj), uj〉Xs 0 (Ω) → 0 〈J ′(uj), u∞〉Xs 0 (Ω) → 0 (4.46) as j → +∞. Thus, by (4.44), (4.45), and (4.46), it is easy to see that ‖uj‖Xs 0 (Ω) → ‖u∞‖Xs 0 (Ω) as j → +∞. This, together with the fact that the sequence {uj}j weakly converges to u∞ in Xs 0(Ω), gives the desired assertion. This concludes the proof of Step 4.4. Step 4.5. The functional J has the Saddle Point Theorem geometry. Let k be as in (4.25) and let us split the space Xs 0(Ω) as Xs 0(Ω) = Hk ⊕ Pk+1. First of all, we show that J is bounded from below in Pk+1. For this purpose, let {uj}j be a sequence in Pk+1 such that ‖uj‖Xs 0 (Ω) → +∞ (4.47) as j → +∞. Arguing as in Step 4.2 and taking into account that (4.25) holds (instead of (4.5)), it is easily seen that lim j→+∞ F (x, uj(x)) ‖uj‖2Xs 0 (Ω) 6 λk+1 2 |v∞(x)|2 (4.48) for a.e. x ∈ Ω, where v∞ ∈ Pk+1 is such that uj/‖uj‖Xs 0 (Ω) → v∞ weakly in Xs 0(Ω) as j → +∞. 16 G. MOLICA BISCI, R. SERVADEI, B. ZHANG EJDE-2022/85 Arguing as in Step 4.2, (4.48) yields that L := lim j→+∞ J (uj) ‖uj‖2Xs 0 (Ω) > 0 . (4.49) Hence, by (4.49), we have J (uj) > L 2 ‖uj‖2Xs 0 (Ω) (4.50) for j sufficiently large. By (4.47) and (4.50) we obtain that J (uj) → +∞ as j → +∞, which yields lim u∈Pk+1, ‖u‖Xs 0(Ω)→+∞ J (u) = +∞ . (4.51) By (4.51) we deduce that there exists M such that J (u) > 1 for all u ∈ Pk+1 with ‖u‖Xs 0 (Ω) >M . (4.52) Now, let u ∈ Pk+1 be such that ‖u‖Xs 0 (Ω) < M . By (2.6) and (4.3) we have J (u) = 1 2 ‖u‖2Xs 0 (Ω) − ∫ Ω F (x, u(x)) dx− ∫ Ω g(x)u(x) dx > −‖a1‖L2(Ω)‖u‖L2(Ω) − 1 2 ‖a2‖L∞(Ω)‖u‖2L2(Ω) − ‖g‖L2(Ω)‖u‖L2(Ω) > − ‖a1‖L2(Ω)√ λ1 M − 1 2 ‖a2‖L∞(Ω) λ1 M2 − ‖g‖L2(Ω)√ λ1 M =: −K . (4.53) By (4.52) and (4.53) we obtain that J (u) > −K for all u ∈ Pk+1, that is J is bounded from below in Pk+1. Now, we have to show that there exists R > 0 such that sup u∈Hk, ‖u‖Xs 0(Ω)=R J (u) < −K. (4.54) where K is given in (4.53). For this purpose, let {uj}j be a sequence in Hk such that ‖uj‖Xs 0 (Ω) → +∞ (4.55) as j → +∞. With the same arguments used for proving (4.48) and taking into account that (2.10) and (4.25) hold, we have lim j→+∞ F (x, uj(x)) ‖uj‖2Xs 0 (Ω) > λk 2 |v∞(x)|2 (4.56) for a.e. x ∈ Ω, with strict inequality when v∞(x) 6= 0. Here v∞ ∈ Hk is such that uj/‖uj‖Xs 0 (Ω) → v∞ weakly in Xs 0(Ω) as j → +∞. Since Hk is a finite dimensional space, ‖v∞‖Xs 0 (Ω) = 1 and so v∞ 6≡ 0 . (4.57) EJDE-2022/85 MONOTONICITY PROPERTIES OF EIGENVALUES 17 Now, by (2.10), (4.55), (4.56), and (4.57), we have L := lim j→+∞ J (uj) ‖uj‖2Xs 0 (Ω) = lim j→+∞ (1 2 − ∫ Ω F (x, uj(x)) ‖uj‖2Xs 0 (Ω) dx− ∫ Ω g(x)uj(x) ‖uj‖2Xs 0 (Ω) dx ) < 1 2 − λk 2 ∫ Ω |v∞(x)|2 dx 6 1 2 − 1 2 ‖v∞‖2Xs 0 (Ω) = 0 , (4.58) since ‖v∞‖Xs 0 (Ω) = 1. By (4.58) we have J (uj) < L 2 ‖uj‖2Xs 0 (Ω) (4.59) for j sufficiently large. By (4.55), (4.58), and (4.59), we obtain that J (uj)→ −∞ as j → +∞. Hence, lim u∈Hk, ‖u‖Xs 0(Ω)→+∞ J (u) = −∞ . Thus, there exists R > 0 such that (4.54) is verified. This concludes the proof of Step 4.5. Thanks to Steps 4.3, 4.4, and 4.5, the functional J satisfies the assumptions of the Saddle Point Theorem. Hence, J admits a critical point and this proves the existence of a weak solution for problem (1.1). Now, it remains to prove that this solution is unique, provided (1.9) is satisfied. For this, let u1, u2 ∈ Xs 0(Ω) be two distinct weak solutions of (4.1). A simple calculation shows that 〈u1 − u2, ϕ〉Xs 0 (Ω) = ∫ Ω ( f(x, u1(x))− f(x, u2(x)) ) ϕ(x)dx = ∫ Ω r(x) ( u1(x)− u2(x) ) ϕ(x)dx for all ϕ ∈ Xs 0(Ω), where r(x) := { f(x,u1(x))−f(x,u2(x)) u1(x)−u2(x) if u1(x) 6= u2(x) 1 2 ( λk + λk+1 ) if u1(x) = u2(x). Note that r is a measurable function and λk < r(x) < λk+1 for a.e. x ∈ Ω by (4.25). Arguing as in Step 4.3 (see the proof of (4.40)), we can show that u1−u2 ≡ 0 in Xs 0(Ω). This is a contradiction and this completes the proof of Theorem 1.1-(ii). � 4.3. Non-existence of solutions. Proof of Theorem 1.1-(iii). We argue by contradiction and we suppose that prob- lem (1.1) admits one weak solution u ∈ Xs 0(Ω). Taking ϕ = e1 as a test function in (4.1), where e1 is the eigenfunction associated to the first eigenvalue λ1 of −LK , we obtain that∫ Ω ( f(x, u(x)) + g(x)− λ1u(x) ) e1(x) dx = 0 . (4.60) 18 G. MOLICA BISCI, R. SERVADEI, B. ZHANG EJDE-2022/85 Taking into account that e1 6≡ 0 and e1 > 0 in Ω, (4.60) is in contradiction to both (1.10) and (1.11). This completes the proof of Theorem 1.1. � We end this section with the following comment. Remark 4.6. Note that in the setting (i) of Theorem 1.1, we need the unique continuation property of the eigenfunctions of −LK corresponding to λ1 just for proving the uniqueness of solution of (1.1). While, in the framework (ii) of Theo- rem 1.1 the same assumption on the eigenfunctions of λk and the ones of λk+1 is necessary both for the proof of the existence and of the uniqueness of weak solution of problem (1.1). 5. Fractional Laplacian problem with asymmetric nonlinearity In this section we consider problem (1.12) and we prove Theorem 1.2. First of all we observe that Theorem 1.2 can not be derived completely by Theorem 1.1. This is so because in the proof of Theorem 1.2 we can not use Proposition 3.2 due to the regularity assumption on the weight r: note that, while in Proposition 3.1 the weight r is a L∞-function, in Proposition 3.2 we require that r ∈ C1(Ω) ∩ L∞(Ω). Proof of Theorem 1.2. For assertion (i) we can use the same arguments of the proof of Theorem 1.1-(i), to prove the existence of at least a weak solution of prob- lem (1.12). We have to make some changes when proving the uniqueness of the solution. For this purpose, assume by contradiction that problem (1.12) admits two dis- tinct weak solutions u1, u2 ∈ Xs 0(Ω) and let v = u1 − u2. Of course v 6≡ 0 and∫ Rn×Rn (v(x)− v(y))(ϕ(x)− ϕ(y)) |x− y|n+2s dx dy = ∫ Ω ( f(x, u1(x))− f(x, u2(x)) ) ϕ(x)dx = ∫ Ω r(x) ( u1(x)− u2(x) ) ϕ(x)dx = ∫ Ω r(x)v(x)ϕ(x)dx (5.1) for all ϕ ∈ Xs 0(Ω), where r(x) := { f(x,u1(x))−f(x,u2(x)) u1(x)−u2(x) if u1(x) 6= u2(x) 0 if u1(x) = u2(x). Note that r is a measurable function not identically zero and, by (1.14), 0 6 r(x) < λ1,s for a.e. x ∈ Ω . (5.2) Testing (5.1) with ϕ = v and taking into account (5.2), we obtain that ‖v‖2Xs 0 (Ω) = ∫ Ω r(x)|v(x)|2 dx < λ1,s‖v‖2L2(Ω) 6 ‖v‖ 2 Xs 0 (Ω) , thanks to the variational characterization of λ1,s. This is a contradiction and this completes the proof of Theorem 1.2-(i). EJDE-2022/85 MONOTONICITY PROPERTIES OF EIGENVALUES 19 Now, let us show Theorem 1.2-(ii). Also in this setting we can argue as in the proof of Theorem 1.1-(ii). We have to change something in the proof of Step 4.3 when proving that (4.40) holds. We have v∞ satisfies∫ Rn×Rn (v∞(x)− v∞(y))(ϕ(x)− ϕ(y)) |x− y|n+2s dx dy = ∫ Ω r(x)v∞(x)ϕ(x)dx (5.3) for all ϕ ∈ Xs 0(Ω), where the weight r is the measurable function given in (4.38), which, by assumption, satisfies the condition λk,s < r(x) < λk+1,s for a.e. x ∈ Ω . (5.4) Since the space Xs 0(Ω) can be split as Xs 0(Ω) = Hk,s ⊕ Pk+1,s, it follows that v∞ can be written as v∞ = v̄∞ + ṽ∞, where v̄∞ ∈ Hk,s and ṽ∞ ∈ Pk+1,s. Taking ϕ = v̄∞ and then ϕ = ṽ∞ as test functions in (5.3) and taking into account the definitions of Hk,s and Pk+1,s, the orthogonality between v̄∞ and ṽ∞ in Xs 0(Ω) and (5.4), we have ‖v̄∞‖2Xs 0 (Ω) = ∫ Ω r(x)|v̄∞(x)|2 dx+ ∫ Ω r(x)v̄∞(x)ṽ∞(x) dx > λk,s‖v̄∞‖2L2(Ω) + ∫ Ω r(x)v̄∞(x)ṽ∞(x) dx (5.5) and ‖ṽ∞‖2Xs 0 (Ω) = ∫ Ω r(x)|ṽ∞(x)|2 dx+ ∫ Ω r(x)ṽ∞(x)v̄∞(x) dx 6 λk+1,s‖ṽ∞‖2L2(Ω) + ∫ Ω r(x)ṽ∞(x)v̄∞(x) dx . (5.6) If v∞ 6≡ 0, then at least one of the functions v̄∞ and ṽ∞ is not identically zero. Thus, at least one of the inequalities (5.5) and (5.6) has to be strict, also thanks to (5.4). Hence, using the variational characterization of the eigenvalues given in (2.9) and (2.10) (with K(x) = |x|−(n+2s)), by (5.5) and (5.6) we have ‖v̄∞‖2Xs 0 (Ω) − ‖ṽ∞‖ 2 Xs 0 (Ω) > λk,s‖v̄∞‖2L2(Ω) − λk+1,s‖ṽ∞‖2L2(Ω) > ‖v̄∞‖2Xs 0 (Ω) − ‖ṽ∞‖ 2 Xs 0 (Ω) , which is a contradiction. This means that v∞ ≡ 0. From here on we can follow the proof of Step 4.3. The remaining part of the proof needs no changes with respect to the proof of Theorem 1.1 and this shows Theorem 1.2. � Acknowledgements. G. Molica Bisci and R. Servadei are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). B. Zhang was supported by National Natural Science Foundation of China (No. 11871199 and No. 12171152). References [1] A. Ambrosetti, D. 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Giovanni Molica Bisci Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino Carlo Bo, Piazza della Repubblica 13, 61029 Urbino (Pesaro e Urbino), Italy Email address: giovanni.molicabisci@uniurb.it Raffaella Servadei Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino Carlo Bo, Piazza della Repubblica 13, 61029 Urbino (Pesaro e Urbino), Italy Email address: raffaella.servadei@uniurb.it Binlin Zhang College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China Email address: zhangbinlin2012@163.com 1. Introduction 2. Preliminaries 2.1. Functional space X0s() and its properties 2.2. Eigenvalues and eigenfunctions of the operator -LK 3. An eigenvalue problem for -LK with weights 3.1. Monotonicity properties of the eigenvalues of nonlocal operators with respect to the weights 4. Variational formulation of the problem and beyond 4.1. Proof of Theorem ??-(i) 4.2. Proof of Theorem ??-(ii) 4.3. Non-existence of solutions 5. Fractional Laplacian problem with asymmetric nonlinearity Acknowledgements References