Multiple concentrating solutions for a fractional Kirchhoff equation with magnetic fields
aa r X i v : . [ m a t h . A P ] O c t MULTIPLE CONCENTRATING SOLUTIONS FOR A FRACTIONAL KIRCHHOFFEQUATION WITH MAGNETIC FIELDS
VINCENZO AMBROSIO
Abstract.
This paper is concerned with the multiplicity and concentration behavior of nontrivial solutionsfor the following fractional Kirchhoff equation in presence of a magnetic field: (cid:0) aε s + bε s − [ u ] A/ε (cid:1) ( − ∆) sA/ε u + V ( x ) u = f ( | u | ) u in R , where ε > is a small parameter, a, b > are constants, s ∈ ( , , ( − ∆) sA is the fractional magneticLaplacian, A : R → R is a smooth magnetic potential, V : R → R is a positive continuous potentialhaving a local minimum and f : R → R is a C subcritical nonlinearity. Applying penalization techniquesand Ljusternik-Schnirelman theory, we relate the number of nontrivial solutions with the topology of theset where the potential V attains its minimum. introduction In this paper, we focus our attention on the following fractional Kirchhoff equation (cid:16) a ε s + b ε s − [ u ] A/ε (cid:17) ( − ∆) sA/ε u + V ( x ) u = f ( | u | ) u in R , (1.1)where ε > is a small parameter, a and b are positive constants, s ∈ ( , , [ u ] A/ε := Z Z R | u ( x ) − e ı ( x − y ) · Aε ( x + y ) u ( y ) | | x − y | s dxdy, the function V : R → R is a continuous potential verifying the following assumptions introduced by delPino and Felmer [18]: ( V ) inf x ∈ R V ( x ) = V > ; ( V ) there exists a bounded domain Λ ⊂ R such that V < min ∂ Λ V and M = { x ∈ Λ : V ( x ) = V } 6 = ∅ , (1.2)and f : R → R is a C -function satisfying the following conditions: ( f ) f ( t ) = 0 for t ≤ and lim t → f ( t ) t = 0 ; ( f ) there exists q ∈ (4 , ∗ s ) , with ∗ s = − s , such that lim t →∞ f ( t ) t q − = 0;( f ) there exists θ ∈ (4 , ∗ s ) such that < θ F ( t ) ≤ tf ( t ) for any t > , where F ( t ) = R t f ( τ ) dτ ; ( f ) there exist σ ∈ (4 , ∗ s ) and C σ > such that f ′ ( t ) t − f ( t ) ≥ C σ t σ − for all t > .We assume that A : R → R is a Hölder continuous magnetic potential of exponent α ∈ (0 , , and ( − ∆) sA is the fractional magnetic Laplacian which, up to a normalization constant, is defined for any u ∈ C ∞ c ( R , C ) as ( − ∆) sA u ( x ) := 2 lim r → Z B cr ( x ) u ( x ) − e ı ( x − y ) · A ( x + y ) u ( y ) | x − y | s dy. This operator has been recently introduced in [17] and relies essentially on the Lévy-Khintchine formulafor the generator of a general Lévy process. For completeness, we emphasize that in the literature there
Mathematics Subject Classification.
Key words and phrases.
Fractional magnetic operators; Kirchhoff equation; variational methods. are three different fractional magnetic operators and that they coincide when s = 1 / and A is assumedto be linear; see [33] for more details.In absence of the magnetic field, i.e. A ≡ , the operator ( − ∆) sA is consistent with the followingdefinition of fractional Laplacian operator ( − ∆) s for smooth functions u ( − ∆) s u ( x ) := 2 lim r → Z B cr ( x ) u ( x ) − u ( y ) | x − y | s dy. This operator arises in a quite natural way in many different physical situations in which one has toconsider long range anomalous diffusions and transport in highly heterogeneous medium; see [19]. When s → , the authors in [44, 52] showed that ( − ∆) sA can be considered as the fractional counterpart of themagnetic Laplacian − ∆ A u := (cid:18) ı ∇ − A (cid:19) u = − ∆ u − ı A ( x ) · ∇ u + | A ( x ) | u − ı u div( A ( x )) , which plays a fundamental role in quantum mechanics in the description of the dynamics of the particlein a non-relativistic setting; see [49]. Motivated by this fact, many authors [1, 11, 15, 21, 36] dealt with theexistence of nontrivial solutions of the following Schrödinger equation with magnetic field − ε ∆ A u + V ( x ) u = f ( x, | u | ) u in R N . (1.3)Equation (1.3) appears when we seek standing wave solutions ψ ( x, t ) = u ( x ) e − ı Eε t , with E ∈ R , for thefollowing time-dependent nonlinear Schrödinger equation with magnetic field: ı ε ∂ψ∂t = (cid:16) εı ∇ − A ( x ) (cid:17) ψ + U ( x ) ψ − f ( | ψ | ) ψ in ( x, t ) ∈ R N × R , where U ( x ) = V ( x ) + E . An important class of solutions of (1.3) are the so called semi-classical stateswhich concentrate and develop a spike shape around one, or more, particular points in R N , while vanishingelsewhere as ε → . This interest is due to the fact that the transition from quantum mechanics to classicalmechanics can be formally performed by sending ε → .Recently, a great attention has been devoted to the study of the following fractional magnetic Schrödingerequation ε s ( − ∆) sA u + V ( x ) u = f ( x, | u | ) u in R N . (1.4)d’Avenia and Squassina [17] studied a class of minimization problems in the spirit of results due to Estebanand Lions in [21]. Fiscella et al. [25] obtained the multiplicity of nontrivial solutions for a fractionalmagnetic problem in a bounded domain. In [7] the author and d’Avenia dealt with the existence andmultiplicity of solutions to (1.4) for small ε > when f has a subcritical growth and the potential V satisfies the following global condition due to Rabinowitz [48]: lim inf | x |→∞ V ( x ) > inf x ∈ R N V ( x ) . (1.5)In [39] Mingqi et al. used suitable variational methods to prove the existence and multiplicity of non-trivial solutions for a class of super-and sub-linear fractional Schrödinger-Kirchhoff equations involving anexternal magnetic potential. We also mention [6, 28, 55] for other interesting results for nonlocal problemsinvolving the operator ( − ∆) sA .We stress that in the case A ≡ , equation (1.1) becomes a fractional Kirchhoff equation of the type (cid:0) a ε s + b ε s − [ u ] (cid:1) ( − ∆) s u + V ( x ) u = f ( x, u ) in R , (1.6)which has been widely studied in the last decade. For instance, when ε = 1 , some existence and multiplicityresults for fractional Kirchhoff equations in R N can be found in [8, 46, 47] and references therein; seealso [22, 26, 27, 42] for problems in bounded domains. In particular, in [9, 31] the authors studied fractionalperturbed Kirchhoff-type problems, that is provided that ε > is sufficiently small. It is worthwhile tomention that Fiscella and Valdinoci [27] proposed for the first time a stationary Kirchhoff model in thefractional setting, which considers the nonlocal aspect of the tension arising from nonlocal measurementsof the fractional length of the string. Such model can be regarded as the nonlocal stationary analogue ofthe Kirchhoff equation ρu tt − (cid:18) p h + E L Z L | u x | dx (cid:19) u xx = 0 , RACTIONAL KIRCHHOFF EQUATIONS WITH MAGNETIC FIELDS 3 which was presented by Kirchhoff [35] in as a generalization of the well-known D’Alembert’s waveequation for free vibrations of elastic strings. The Kirchhoff’s model takes into account the changesin length of the string produced by transverse vibrations. Here u = u ( x, t ) is the transverse stringdisplacement at the space coordinate x and time t , L is the length of the string, h is the area of thecross section, E is Young’s modulus of the material, ρ is the mass density, and p is the initial tension;see [13, 38, 45]. In the classical framework, probably the first result concerning the following perturbedKirchhoff equation − (cid:18) a ε + b ε Z R |∇ u | dx (cid:19) ∆ u + V ( x ) u = f ( u ) in R , (1.7)has been obtained by He and Zou [30], who proved the multiplicity and concentration behavior of positivesolutions to (1.7) for ε > small, under assumption (1.5) on V and involving a subcritical nonlinearity.Subsequently, Wang et al. [53] investigated the multiplicity and concentration phenomenon for (1.7) inpresence of a critical term. Under local conditions ( V ) - ( V ) , Figueiredo and Santos Junior [23] proved amultiplicity result for a subcritical Kirchhoff equation via the generalized Nehari manifold method. Theexistence and concentration of positive solutions for (1.7) with critical growth, has been considered in [29].On the other hand, when we take b = 0 in (1.6), then one has the following fractional Schrödingerequation (see [37]) ε s ( − ∆) s u + V ( x ) u = f ( x, u ) in R N , (1.8)for which several existence and multiplicity results under different assumptions on V and f have beenestablished via appropriate variational and topological methods; see [4, 5, 16, 20, 50] and references therein.In particular way, Davila et al. [16] proved that if V ∈ C ,α ( R N ) ∩ L ∞ ( R N ) and inf x ∈ R N V ( x ) > ,then (1.1) has multi-peak solutions. Alves and Miyagaki [2] (see also [4, 5]) considered the existence andconcentration of positive solutions of (1.8) when V satisfies ( V ) - ( V ) and f has a subcritical growth.Recently, the author and Isernia [10] studied the multiplicity and concentration of positive solutions for afractional Schrödinger equation involving the fractional p -Laplacian operator when the potential satisfies(1.5) and the nonlinearity is assumed to be subcritical or critical.Particularly motivated by the above works and the interest shared by the mathematical community onnonlocal magnetic problems, in this paper we deal with the multiplicity and concentration of nontrivialsolutions to (1.1) when ε → , under assumptions ( V ) - ( V ) and ( f ) - ( f ) . More precisely, our main resultis the following one: Theorem 1.1.
Assume that ( V ) - ( V ) and ( f ) - ( f ) hold. Then, for any δ > such that M δ = { x ∈ R : dist ( x, M ) ≤ δ } ⊂ Λ , there exists ε δ > such that, for any ε ∈ (0 , ε δ ) , problem (1.1) has at least cat M δ ( M ) nontrivial solutions.Moreover, if u ε denotes one of these solutions and x ε is a global maximum point of | u ε | , then we have lim ε → V ( x ε ) = V and | u ε ( x ) | ≤ C ε s C ε s + | x − x ε | s ∀ x ∈ R . In what follows we give a sketch of the proof of Theorem 1.1. Firstly, using the change of variable x ε x , instead of (1.1), we can consider the following equivalent problem (cid:0) a + b [ u ] A ε (cid:1) ( − ∆) sA ε u + V ε ( x ) u = f ( | u | ) u in R , (1.9)where A ε ( x ) := A ( ε x ) and V ε ( x ) := V ( ε x ) . Due to the lack of information on the behavior of V atinfinity, inspired by [1, 18], we modify the nonlinearity f in an appropriate way, considering an auxiliaryproblem. In this way, we are able to apply suitable variational arguments to study the modified problem,and then we prove that, for ε > small enough, the solutions of the modified problem are also solutionsof the original one. More precisely, we fix k > and a ′ > such that f ( a ′ ) = V k , and we consider thefunction ˆ f ( t ) := ( f ( t ) if t ≤ a ′ V k if t > a ′ . Let t a ′ , T a ′ > such that t a ′ < a ′ < T a ′ and take ξ ∈ C ∞ c ( R , R ) such that: V. AMBROSIO ( ξ ) ξ ( t ) ≤ ˆ f ( t ) for all t ∈ [ t a ′ , T a ′ ] , ( ξ ) ξ ( t a ′ ) = ˆ f ( t a ′ ) , ξ ( T a ′ ) = ˆ f ( T a ′ ) , ξ ′ ( t a ′ ) = ˆ f ′ ( t a ′ ) and ξ ′ ( T a ′ ) = ˆ f ′ ( T a ′ ) , ( ξ ) the map t ξ ( t ) is increasing for all t ∈ [ t a ′ , T a ′ ] .Then we define ˜ f ∈ C ( R , R ) as follows: ˜ f ( t ) := ( ˆ f ( t ) if t / ∈ [ t a ′ , T a ′ ] ξ ( t ) if t ∈ [ t a ′ , T a ′ ] . Finally, we introduce the following penalized nonlinearity g : R × R → R by setting g ( x, t ) = χ Λ ( x ) f ( t ) + (1 − χ Λ ( x )) ˜ f ( t ) , where χ Λ is the characteristic function on Λ , and we set G ( x, t ) = R t g ( x, τ ) dτ . From assumptions ( f ) - ( f ) and ( ξ ) - ( ξ ) , it follows that g verifies the following properties:( g ) lim t → g ( x, t ) t = 0 uniformly in x ∈ R ;( g ) g ( x, t ) ≤ f ( t ) for any x ∈ R and t > ;( g ) ( i ) 0 < θ G ( x, t ) ≤ g ( x, t ) t for any x ∈ Λ and t > , ( ii ) 0 ≤ G ( x, t ) ≤ g ( x, t ) t ≤ V ( x ) k t and ≤ g ( x, t ) ≤ V ( x ) k for any x ∈ Λ c and t > ;( g ) t g ( x,t ) t is increasing for all x ∈ Λ and t > .Then we introduce the following modified problem (cid:0) a + b [ u ] A ε (cid:1) ( − ∆) sA ε u + V ε ( x ) u = g ε ( x, | u | ) u in R . (1.10)Let us note that if u is a solution of (1.10) such that | u ( x ) | ≤ t a ′ for all x ∈ Λ cε , (1.11)where Λ ε := { x ∈ R N : ε x ∈ Λ } , then u is also a solution of (1.9). Therefore, in order to study weaksolutions of (1.10), we look for critical points of the following functional associated with (1.9): J ε ( u ) = a u ] A ε + 12 Z R V ε ( x ) | u | dx + b u ] A ε − Z R G ( ε x, | u | ) dx defined on the fractional magnetic Sobolev space H sε = (cid:26) u ∈ D sA ε ( R , C ) : Z R V ε ( x ) | u | dx < ∞ (cid:27) ; see Section . From ( g ) - ( g ) , it is easy to check that J ε has mountain pass geometry [3]. Anyway, thepresence of the magnetic field and the lack of compactness of the embeddings H sε into L p ( R , R ) , with p ∈ (2 , ∗ s ) , create several difficulties to show that J ε verifies the Palais-Smale condition ( ( P S ) in short).More precisely, the Kirchhoff term [ u ] A ( − ∆) sA does not permit to deduce in standard way that weak limitsof (bounded) Palais-Smale sequences of J ε are critical points of it. Therefore, a more careful investigationwill be needed to recover some compactness property for the modified functional. After that, combiningsome ideas introduced by Benci and Cerami [12] with the Ljusternik-Schnirelman theory, we deduce amultiplicity result for the modified problem. We point out that the Hölder regularity assumption on themagnetic field A and the fractional diamagnetic inequality [17], will play a very important role to apply theminimax methods; see Sections and . In order to show that the solutions of (1.10) are indeed solutionsof (1.9), we need to show that (1.11) holds for ε small enough. This property will be proved using asuitable variant of the Moser iteration argument [41] and a sort of Kato’s inequality [34] for ( − ∆) sA . Westress that L ∞ -estimates as in [1] seem very hard to adapt in the nonlocal magnetic framework. Moreover,differently from the classical magnetic case (see [15, 36]), we do not have a Kato’s inequality for ( − ∆) sA (except for s = 1 / as showed in [32], while here we are assuming s > / ). Therefore, in this work wedevelop some new ingredients which we believe to be useful for future problems like (1.1). We also providea decay estimate of solutions of (1.1) which is in clear accordance with the results in [24]. As far as weknow, this is the first time that penalization methods jointly with Ljusternik-Schnirelmann theory areused to obtain multiple solutions for a fractional Kirchhoff equation with magnetic fields. RACTIONAL KIRCHHOFF EQUATIONS WITH MAGNETIC FIELDS 5
We organize the paper in the following way: in Section we collect some preliminary results for fractionalSobolev spaces; in Section we study the modified functional; in Section we provide a multiplicity resultfor (1.10); finally, in Section , we present the proof of Theorem 1.1.2. Preliminaries
In this preliminary section we fix the notations and we recall some technical results. We denote by H s ( R , R ) the fractional Sobolev space H s ( R , R ) = { u ∈ L ( R , R ) : [ u ] < ∞} , where [ u ] = Z Z R | u ( x ) − u ( y ) | | x − y | s dxdy is the Gagliardo seminorm. We recall that the embedding H s ( R , R ) ⊂ L q ( R , R ) is continuous for all q ∈ [2 , ∗ s ) and locally compact for all q ∈ [1 , ∗ s ) ; see [19, 40].Let L ( R , C ) be the space of complex-valued functions such that k u k L ( R ) = R R | u | dx < ∞ endowedwith the inner product h u, v i L = ℜ R R u ¯ v dx , where the bar denotes complex conjugation.Let us denote by [ u ] A := Z Z R | u ( x ) − e ı ( x − y ) · A ( x + y ) u ( y ) | | x − y | s dxdy, and consider D sA ( R , C ) := n u ∈ L ∗ s ( R , C ) : [ u ] A < ∞ o . Then, we consider the Hilbert space H sε := (cid:26) u ∈ D sA ε ( R , C ) : Z R V ε ( x ) | u | dx < ∞ (cid:27) endowed with the scalar product h u, v i ε := a ℜ Z Z R ( u ( x ) − e ı ( x − y ) · A ε ( x + y ) u ( y ))( v ( x ) − e ı ( x − y ) · A ε ( x + y ) v ( y )) | x − y | s dxdy + ℜ Z R V ε ( x ) u ¯ vdx for all u, v ∈ H sε , and let k u k ε := p h u, u i ε . In what follows we list some useful lemmas; see [7, 17] for more details.
Lemma 2.1. [7, 17] The space H sε is complete and C ∞ c ( R , C ) is dense in H sε . Lemma 2.2. [17] If u ∈ H sA ( R , C ) then | u | ∈ H s ( R , R ) and we have [ | u | ] ≤ [ u ] A . Theorem 2.1. [17] The space H sε is continuously embedded in L r ( R , C ) for r ∈ [2 , ∗ s ] , and compactlyembedded in L r loc ( R , C ) for r ∈ [1 , ∗ s ) . Lemma 2.3. [7] If u ∈ H s ( R , R ) and u has compact support, then w = e ıA (0) · x u ∈ H sε . We also recall a fractional version of Lions lemma whose proof can be found in [24]:
Lemma 2.4. [24] Let q ∈ [2 , ∗ s ) . If ( u n ) is a bounded sequence in H s ( R , R ) and if lim n →∞ sup y ∈ R Z B R ( y ) | u n | q dx = 0 for some R > , then u n → in L r ( R , R ) for all r ∈ (2 , ∗ s ) . V. AMBROSIO variational setting and the modified functional Let us introduce the following functional J ε : H sε → R defined as J ε ( u ) = 12 k u k ε + b u ] A ε − Z R G ( ε x, | u | ) dx. It is easy to check that J ε ∈ C ( H sε , R ) and that its differential J ′ ε is given by h J ′ ε ( u ) , v i = h u, v i ε + b [ u ] A ε ℜ Z Z R ( u ( x ) − e ı ( x − y ) · A ε ( x + y ) u ( y ))( v ( x ) − e ı ( x − y ) · A ε ( x + y ) v ( y )) | x − y | s dxdy − ℜ Z R g ( ε x, | u | ) u ¯ vdx. Therefore, weak solutions to (1.10) can be found as critical points of J ε . We will also consider the followingfamily of autonomous problems associated to (1.10), that is for all µ > a + b [ u ] )( − ∆) s u + µu = f ( u ) u in R , (3.1)and we introduce the corresponding energy functional J µ : H sµ → R given by J µ ( u ) = 12 k u k µ + b u ] − Z R F ( u ) dx where H sµ stands for the fractional Sobolev space H s ( R , R ) endowed with the norm k u k µ = a [ u ] + µ k u k L ( R ) . We stress that, under the assumptions on f , J ε possesses a mountain pass geometry [3]. Indeed, we canprove that: Lemma 3.1. ( i ) J ε (0) = 0 ; ( ii ) there exists α, ρ > such that J ε ( u ) ≥ α for any u ∈ H sε such that k u k ε = ρ ; ( iii ) there exists e ∈ H sε with k e k ε > ρ such that J ε ( e ) < .Proof. By ( g ) and ( g ) , for all δ > there exists C δ > such that | G ( ε x, t ) | ≤ δ | t | + C δ | t | q for all x ∈ R , t ∈ R . This fact combined with Theorem 2.1 implies that J ε ( u ) ≥ C k u k ε − δC k u k ε − C δ k u k qε . Since q ∈ (4 , ∗ s ) , it follows that ( i ) holds. Now, fix u ∈ H sε \ { } with supp ( u ) ⊂ Λ ε . By ( f ) we get J ε ( T u ) ≤ T k u k ε + b T u ] A ε − Z Λ ε F ( T | u | ) dx ≤ T k u k ε + T b [ u ] A ε − CT θ Z Λ ε | u | θ dx + C which in view of θ > yields J ε ( T u ) → −∞ as T → ∞ . (cid:3) From Lemma 3.1 it follows that we can define the minimax level c ε = inf γ ∈ Γ ε max t ∈ [0 , J ε ( γ ( t )) where Γ ε = { γ ∈ C ([0 , , H sε ) : γ (0) = 0 and J ε ( γ (1)) < } . Using a version of the mountain pass theorem without ( P S ) condition (see [54]), we can find a Palais-Smalesequence ( u n ) at the level c ε . Now, we prove that J ε enjoys of the following compactness property: Lemma 3.2.
Let c ∈ R . Then J ε satisfies the Palais-Smale condition at the level c . RACTIONAL KIRCHHOFF EQUATIONS WITH MAGNETIC FIELDS 7
Proof.
Let ( u n ) ⊂ H sε be a ( P S ) c -sequence of J ε , that is J ε ( u n ) → c and J ′ ε ( u n ) → as n → ∞ . Sincethe proof is very long, we divide it into four steps. Step 1
The sequence ( u n ) is bounded in H sε . Indeed, by ( g ) we get c + o n (1) k u n k ε = J ε ( u n ) − θ h J ′ ε ( u n ) , u n i = (cid:18) − θ (cid:19) k u n k ε + (cid:18) − θ (cid:19) b [ u n ] A ε + 1 θ Z R (cid:20) g ε ( x, | u n | ) | u n | − θ G ε ( x, | u n | ) (cid:21) dx ≥ (cid:18) − θ (cid:19) k u n k ε + (cid:18) − θ θ (cid:19) Z Λ cε G ε ( x, | u n | ) dx ≥ (cid:18) − θ (cid:19) k u n k ε + (cid:18) − θ θk (cid:19) Z Λ cε V ( ε x ) | u n | dx ≥ (cid:18) θ − θ (cid:19) (cid:18) − k (cid:19) k u n k ε , and using the fact that k > , we can conclude that ( u n ) is bounded in H sε . Consequently, we may assumethat u n ⇀ u in H sε and [ u n ] A ε → ℓ ∈ (0 , ∞ ) .Set M n := a + b [ u n ] A ε , and we note that M n → a + bℓ as n → ∞ . In what follows we prove that u n → u in H sε . Step 2
For any ξ > there exists R = R ξ > such that Λ ε ⊂ B R and lim sup n →∞ Z B cR Z R a | u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) | | x − y | s dxdy + Z B cR V ε ( x ) | u n ( x ) | dx ≤ ξ. (3.2)Let η R ∈ C ∞ ( R , R ) be such that ≤ η R ≤ , η R = 0 in B R , η R = 1 in B cR and |∇ η R | ≤ CR for some C > independent of R . From h J ′ ε ( u n ) , η R u n i = o n (1) it follows that ℜ M n Z Z R ( u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) )( u n ( x ) η R ( x ) − u n ( y ) η R ( y ) e ıA ε ( x + y ) · ( x − y ) ) | x − y | s dxdy + Z R V ε η R | u n | dx = Z R N g ε ( x, | u n | ) | u n | η R dx + o n (1) . Since ℜ Z Z R ( u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) )( u n ( x ) η R ( x ) − u n ( y ) η R ( y ) e ıA ε ( x + y ) · ( x − y ) ) | x − y | s dxdy = ℜ Z Z R u n ( y ) e − ıA ε ( x + y ) · ( x − y ) ( u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) )( η R ( x ) − η R ( y )) | x − y | s dxdy ! + Z Z R η R ( x ) | u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) | | x − y | s dxdy, and M n ≥ a , we can use ( g ) -(ii) to get a Z Z R η R ( x ) | u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) | | x − y | s dxdy + Z R V ε ( x ) η R | u n | dx ≤ −ℜ " M n Z Z R u n ( y ) e − ıA ε ( x + y ) · ( x − y ) ( u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) )( η R ( x ) − η R ( y )) | x − y | s dxdy + 1 k Z R V ε η R | u n | dx + o n (1) . (3.3) V. AMBROSIO
Using Hölder inequality and the boundedness of ( u n ) in H sε we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℜ " M n Z Z R u n ( y ) e − ıA ε ( x + y ) · ( x − y ) ( u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) )( η R ( x ) − η R ( y )) | x − y | s dxdy ≤ C Z Z R | u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) | | x − y | s dxdy ! (cid:18)Z Z R | u n ( y ) | | η R ( x ) − η R ( y ) | | x − y | s dxdy (cid:19) ≤ C (cid:18)Z Z R | u n ( y ) | | η R ( x ) − η R ( y ) | | x − y | s dxdy (cid:19) . (3.4)Now, we show that lim sup R →∞ lim sup n →∞ Z Z R | u n ( y ) | | η R ( x ) − η R ( y ) | | x − y | s dxdy = 0 . (3.5)Let us note that R = (( R \ B R ) × ( R \ B R )) ∪ (( R \ B R ) × B R ) ∪ ( B R × R ) =: X R ∪ X R ∪ X R . Therefore
Z Z R | η R ( x ) − η R ( y ) | | x − y | s | u n ( x ) | dxdy = Z Z X R | η R ( x ) − η R ( y ) | | x − y | s | u n ( x ) | dxdy + Z Z X R | η R ( x ) − η R ( y ) | | x − y | s | u n ( x ) | dxdy + Z Z X R | η R ( x ) − η R ( y ) | | x − y | s | u n ( x ) | dxdy. (3.6)Since η R = 1 in R \ B R , we can see that Z Z X R | u n ( x ) | | η R ( x ) − η R ( y ) | | x − y | s dxdy = 0 . (3.7)Now, fix K > , and we observe that X R = ( R \ B R ) × B R ⊂ (( R \ B KR ) × B R ) ∪ (( B KR \ B R ) × B R ) If ( x, y ) ∈ ( R \ B KR ) × B R , then | x − y | ≥ | x | − | y | ≥ | x | − R > | x | . Therefore, using ≤ η R ≤ , |∇ η R | ≤ CR and applying Hölder inequality we obtain Z Z X R | u n ( x ) | | η R ( x ) − η R ( y ) | | x − y | s dxdy = Z R \ B KR Z B R | u n ( x ) | | η R ( x ) − η R ( y ) | | x − y | s dxdy + Z B kR \ B R Z B R | u n ( x ) | | η R ( x ) − η R ( y ) | | x − y | s dxdy ≤ s Z R \ B KR Z B R | u n ( x ) | | x | s dxdy + CR Z B KR \ B R Z B R | u n ( x ) | | x − y | s − dxdy ≤ CR Z R \ B KR | u n ( x ) | | x | s dx + CR ( KR ) − s ) Z B KR \ B R | u n ( x ) | dx ≤ CR Z R \ B KR | u n ( x ) | ∗ s dx ! ∗ s Z R \ B KR | x | s +3 dx ! s + CK − s ) R s Z B KR \ B R | u n ( x ) | dx ≤ CK Z R \ B KR | u n ( x ) | ∗ s dx ! ∗ s + CK − s ) R s Z B KR \ B R | u n ( x ) | dx ≤ CK + CK − s ) R s Z B KR \ B R | u n ( x ) | dx. (3.8) RACTIONAL KIRCHHOFF EQUATIONS WITH MAGNETIC FIELDS 9
Take δ ∈ (0 , , and we obtain Z Z X R | u n ( x ) | | η R ( x ) − η R ( y ) | | x − y | s dxdy ≤ Z B R \ B δR Z R | u n ( x ) | | η R ( x ) − η R ( y ) | | x − y | s dxdy + Z B δR Z R | u n ( x ) | | η R ( x ) − η R ( y ) | | x − y | s dxdy. (3.9)Since Z B R \ B δR Z R ∩{ y : | x − y |
For all
R > it holds lim n →∞ Z B R dx Z R | u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) | | x − y | s dy + Z B R V ε | u n | dx = Z B R dx Z R | u ( x ) − u ( y ) e ıA ε ( x + y ) · ( x − y ) | | x − y | s dy + Z B R V ε | u | dx. (3.14)Let η ρ ∈ C ∞ ( R , R ) be such that η ρ = 1 in B ρ and η ρ = 0 in B c ρ , with ≤ η ρ ≤ .Set Φ n ( x ) := M n Z R | ( u n ( x ) − u ( x )) − ( u n ( y ) − u ( y )) e ıA ε ( x + y ) · ( x − y ) ) | | x − y | s dy + V ε | u n ( x ) − u ( x ) | . RACTIONAL KIRCHHOFF EQUATIONS WITH MAGNETIC FIELDS 11
Fix
R > and choose ρ > R . Then we have ≤ Z B R Φ n ( x ) dx = Z B R Φ n ( x ) η ρ ( x ) dx ≤ M n Z Z R | ( u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) ) − ( u ( x ) − u ( y ) e ıA ε ( x + y ) · ( x − y ) ) | | x − y | s η ρ ( x ) dxdy + Z R V ε | u n − u | η ρ dx = M n Z Z R | u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) | | x − y | s η ρ ( x ) dxdy + Z R V ε | u n | η ρ dx + M n Z Z R | u ( x ) − u ( y ) e ıA ε ( x + y ) · ( x − y ) | | x − y | s η ρ ( x ) dxdy + Z R V ε | u | η ρ dx − ℜ M n Z Z R ( u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) )( u ( x ) − u ( y ) e ıA ε ( x + y ) · ( x − y ) ) | x − y | s η ρ ( x ) dxdy + Z R V ε u n ¯ u η ρ dx (cid:21) = I n,ρ − II n,ρ + III n,ρ + IV n,ρ ≤ | I n,ρ | + | II n,ρ | + | III n,ρ | + | IV n,ρ | , (3.15)where I n,ρ := M n Z Z R | u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) | | x − y | s η ρ ( x ) dxdy + Z R V ε | u n | η ρ dx − Z R g ( ε x, | u n | ) | u n | η ρ dx,II n,ρ := ℜ M n Z Z R ( u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) )( u ( x ) − u ( y ) e ıA ε ( x + y ) · ( x − y ) ) | x − y | s η ρ ( x ) dxdy + Z R V ε u n ¯ uη ρ dx (cid:21) − ℜ Z R g ( ε x, | u n | ) u n ¯ uη ρ dx,III n,ρ := −ℜ M n Z Z R ( u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) )( u ( x ) − u ( y ) e ıA ε ( x + y ) · ( x − y ) ) | x − y | s η ρ ( x ) dxdy + Z R V ε u n ¯ uη ρ dx (cid:21) + M n Z Z R | u ( x ) − u ( y ) e ıA ε ( x + y ) · ( x − y ) | | x − y | s η ρ ( x ) dxdy + Z R V ε | u | η ρ dx =: − III n,ρ + III n,ρ ,IV n,ρ := Z R g ( ε x, | u n | ) | u n | η ρ dx − ℜ Z R g ( ε x, | u n | ) u n ¯ uη ρ dx. Let us prove that lim ρ →∞ lim sup n →∞ | I n,ρ | = 0 . (3.16)Firstly, we note that I n,ρ can be written as I n,ρ = h J ′ ε ( u n ) , u n η ρ i− ℜ " M n Z Z R ( u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) )( η ρ ( x ) − η ρ ( y )) | x − y | s u n ( y ) e − ıA ε ( x + y ) · ( x − y ) dxdy . Since ( u n η ρ ) is bounded in H sε , we have h J ′ ε ( u n ) , u n η ρ i = o n (1) , and then I n,ρ = o n (1) − ℜ " M n Z Z R ( u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) )( η ρ ( x ) − η ρ ( y )) | x − y | s u n ( y ) e − ıA ε ( x + y ) · ( x − y ) dxdy . (3.17) Applying Hölder inequality and using the boundedness of ( u n ) in H sε and (3.5) with η R = 1 − η ρ , we caninfer that lim ρ →∞ lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M n Z Z R ( u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) )( η ρ ( x ) − η ρ ( y )) | x − y | s u n ( y ) e − ıA ε ( x + y ) · ( x − y ) dxdy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , which together with (3.17) yields (3.16). Now, we note that II n,ρ = h J ′ ε ( u n ) , uη ρ i− ℜ " M n Z Z R ( u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) )( η ρ ( x ) − η ρ ( y )) | x − y | s u ( y ) e − ıA ε ( x + y ) · ( x − y ) dxdy . Proceeding as in the previous case, we can show that lim ρ →∞ lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M n Z Z R ( u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) )( η ρ ( x ) − η ρ ( y )) | x − y | s u ( y ) e − ıA ε ( x + y ) · ( x − y ) dxdy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , and being h J ′ ε ( u n ) , uη ρ i = o n (1) , we obtain lim ρ →∞ lim sup n →∞ | II n,ρ | = 0 . (3.18)Now, we show that lim ρ →∞ lim n →∞ | III n,ρ | = 0 . (3.19)Firstly, we can use M n → a + bℓ and the Dominated Convergence Theorem to see that lim ρ →∞ lim n →∞ III n,ρ = ( a + bℓ )[ u ] A ε + Z R V ε | u | dx =: L. (3.20)On the other hand, we can observe that III n,ρ can be written as follows: III n,ρ = ℜ M n Z Z R ( u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) )( u ( x ) η ρ ( x ) − u ( y ) η ρ ( y ) e ıA ε ( x + y ) · ( x − y ) ) | x − y | s dxdy + ℜ Z R V ε u n ¯ uη ρ dx − ℜ " M n Z Z R ( u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) )( η ρ ( x ) − η ρ ( y )) | x − y | s u ( y ) e − ıA ε ( x + y ) · ( x − y ) dxdy =: A n,ρ − B n,ρ . (3.21)From the weak convergence of ( u n ) and M n → a + bℓ we can obtain that lim n →∞ A n,ρ = ℜ ( a + bℓ ) Z Z R ( u ( x ) − u ( y ) e ıA ε ( x + y ) · ( x − y ) )( u ( x ) η ρ ( x ) − u ( y ) η ρ ( y ) e ıA ε ( x + y ) · ( x − y ) ) | x − y | s dxdy + ℜ Z R V ε | u | η ρ dx = ℜ " ( a + bℓ ) Z Z R | u ( x ) − u ( y ) e ıA ε ( x + y ) · ( x − y ) | | x − y | s η ρ ( x ) dxdy + ℜ " ( a + bℓ ) Z Z R ( u ( x ) − u ( y ) e ıA ε ( x + y ) · ( x − y ) )( η ρ ( x ) − η ρ ( y )) | x − y | s u ( y ) e − ıA ε ( x + y ) · ( x − y ) dxdy + ℜ Z R V ε | u | η ρ dx. RACTIONAL KIRCHHOFF EQUATIONS WITH MAGNETIC FIELDS 13
Noting that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Z R ( u ( x ) − u ( y ) e ıA ε ( x + y ) · ( x − y ) )( η ρ ( x ) − η ρ ( y )) | x − y | s u ( y ) e − ıA ε ( x + y ) · ( x − y ) dxdy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ [ u ] A ε (cid:18)Z Z R | u ( y ) | | η ρ ( x ) − η ρ ( y ) | | x − y | s dxdy (cid:19) / → as ρ → ∞ (3.22)(one can argue as in (3.5)), and using the Dominated Convergence Theorem we can deduce that lim ρ →∞ lim n →∞ A n,ρ = L. Similarly to (3.22), we also have lim sup n →∞ | B n,ρ | ≤ C (cid:18)Z Z R | u ( y ) | | η ρ ( x ) − η ρ ( y ) | | x − y | s dxdy (cid:19) / → as ρ → ∞ . From the above relations of limits we can infer that lim ρ →∞ lim n →∞ III n,ρ = L. (3.23)Combining (3.20) and (3.23) and using the definition of III n,ρ we can conclude that (3.19) holds true.In the light of ( g ) and ( g ) and the strong convergence of | u n | → | u | in L ploc ( R , R ) for ≤ p < − s (by Theorem 2.1), we deduce that for any ρ > R it holds lim n →∞ | IV n,ρ | = 0 . (3.24)Putting together (3.15), (3.16), (3.18), (3.19) and (3.24) we get ≤ lim sup n →∞ Z B R Φ n ( x ) dx ≤ , that is lim n →∞ R B R Φ n ( x ) dx = 0 which yields (3.14). Step 4
Conclusion. Using (3.2) we know that for each ζ > there exists R = R ( ζ ) > Cζ such that lim sup n →∞ "Z R \ B R dx Z R a | u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) | | x − y | s dy + Z R \ B R V ( ε x ) | u n | dx < ζ. (3.25)Taking into account u n ⇀ u in H sε , (3.25) and (3.14) we can infer k u k ε ≤ lim inf n →∞ k u n k ε ≤ lim sup n →∞ k u n k ε = lim sup n →∞ h Z B R dx Z R a | u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) | | x − y | s dy + Z B R V ε | u n | dx + Z R \ B R dx Z R a | u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) | | x − y | s dy + Z R \ B R V ε | u n | dx i ≤ Z B R dx Z R a | u ( x ) − u ( y ) e ıA ε ( x + y ) · ( x − y ) | | x − y | s dy + Z B R V ε | u | dx + ζ. Since R → ∞ as ζ → , we get k u k ε ≤ lim inf n →∞ k u n k ε ≤ lim sup n →∞ k u n k ε ≤ k u k ε , which implies k u n k ε → k u k ε . Recalling that H sε is a Hilbert space, we can deduce that u n → u in H sε as n → ∞ . (cid:3) Since we are looking for multiple critical points of the functional J ε , we shall consider it constrained toan appropriated subset of H sε . More precisely, we define the Nehari manifold associated to (1.10), that is N ε := { u ∈ H sε \ { } : h J ′ ε ( u ) , u i = 0 } , and we indicate by N µ the Nehari manifold associated to (3.1). Moreover, it is easy to show (see [54])that c ε can be characterized as follows: c ε = inf u ∈ H sε \{ } sup t ≥ J ε ( tu ) = inf u ∈N ε J ε ( u ) . In what follows, we denote by c µ the minimax level for the autonomous problem (3.1).From the growth conditions of g , we can see that for a fixed u ∈ N ε ≥ k u k ε − Z R g ( ε n x, | u | ) | u | dx ≥ k u k ε − k Z R V ε ( x ) | u | dx − C k u k qε ≥ min (cid:26) a, k − k (cid:27) k u k ε − C k u k qε , so there exists r > independent of u such that k u k ε ≥ r for all u ∈ N ε . (3.26)Now, we prove the following result. Proposition 3.1.
Let c ∈ R . Then, the functional J ε restricted to N ε satisfies the ( P S ) c condition at thelevel c .Proof. Let ( u n ) ⊂ N ε be such that J ε ( u n ) → c and k J ′ ε ( u n ) |N ε k ∗ = o n (1) . Then (see [54]) we can find ( λ n ) ⊂ R such that J ′ ε ( u n ) = λ n T ′ ε ( u n ) + o n (1) , (3.27)where T ε : H sε → R is defined as T ε ( u ) = k u k ε + b [ u ] A ε − Z R g ( ε x, | u | ) | u | dx. In view of h J ′ ε ( u n ) , u n i = 0 , g ( ε x, | u | ) is constant in Λ cε ∩ {| u | > T a ′ } , and using the definitions of g , themonotonicity of η and ( f ) , we obtain h T ′ ε ( u n ) , u n i = 2 k u n k ε + 4 b [ u n ] A ε − Z R g ′ ( ε x, | u n | ) | u n | dx − Z R g ( ε x, | u n | ) | u n | dx = − k u n k ε + 2 Z R g ( ε x, | u n | ) | u n | dx − Z R g ′ ( ε x, | u n | ) | u n | dx ≤ − C Z Λ ε ∪{| u n | Corollary 3.1. The critical points of the functional J ε on N ε are critical points of J ε . At this point, we provide some useful results about Kirchhoff autonomous problems (3.1). We beginproving the following Lions compactness result. RACTIONAL KIRCHHOFF EQUATIONS WITH MAGNETIC FIELDS 15 Lemma 3.3. Let ( u n ) ⊂ H sµ be a ( P S ) c sequence for J µ . Then one of the following conclusions holds: ( i ) u n → in H sµ ; ( ii ) there exists a sequence ( y n ) ⊂ R and constants R, β > such that lim inf n →∞ Z B R ( y n ) | u n | dx ≥ β > . Proof. Assume that ( ii ) does not occur. Arguing as in the proof of Lemma 3.2 we can see that ( u n ) isbounded in H sµ . Then we can use Lemma 2.4 to deduce that u n → in L r ( R , R ) for all r ∈ (2 , ∗ s ) .In view of ( f ) - ( f ) we get R R f ( u n ) u n dx = o n (1) . This fact combined with h J ′ µ ( u n ) , u n i = o n (1) yields k u n k µ ≤ k u n k µ + b [ u n ] = R R f ( u n ) u n dx + o n (1) = o n (1) . (cid:3) Therefore, we can prove an existence result for the autonomous Kirchhoff problem. Lemma 3.4. Fo all µ > , there exists a positive ground state solution of (3.1) .Proof. It is easy to check that J µ has a mountain pass geometry, so there exists a sequence ( u n ) ⊂ H sµ suchthat J µ ( u n ) → c µ and J ′ µ ( u n ) → . Thus, ( u n ) is bounded in H sµ and we may assume that u n ⇀ u in H sµ and [ u n ] → B . Suppose that u = 0 . Since h J ′ µ ( u n ) , ϕ i = o n (1) we can deduce that for all ϕ ∈ C ∞ c ( R , R ) Z R a ( − ∆) s u ( − ∆) s ϕ + µuϕ dx + bB (cid:18)Z R ( − ∆) s u ( − ∆) s ϕ dx (cid:19) − Z R f ( u ) uϕ dx = 0 . (3.29)Let us note that B ≥ [ u ] by Fatou’s Lemma. If by contradiction B > [ u ] , we may use (3.29) to deducethat h J ′ µ ( u ) , u i < . Moreover, conditions ( f ) - ( f ) imply that h J ′ µ ( tu ) , tu i > for small t > . Thenthere exists t ∈ (0 , such that t u ∈ N µ and h J ′ µ ( t u ) , t u i = 0 . Using Fatou’s Lemma, t ∈ (0 , and f ( t ) t − F ( t ) is increasing for t > (by ( f ) and ( f ) ) we get c µ ≤ J µ ( t u ) − h J ′ µ ( t u ) , t u i < lim inf n →∞ (cid:20) J µ ( u n ) − h J ′ µ ( u n ) , u n i (cid:21) = c µ which gives a contradiction. Therefore B = [ u ] and we deduce that J ′ µ ( u ) = 0 . Hence u ∈ N µ . Usingthe fact that h J ′ µ ( u ) , u − i = 0 and ( f ) we can see that u ≥ in R . Moreover we can argue as in Lemma5.1 to infer that u ∈ L ∞ ( R , R ) . Since u satisfies ( − ∆) s u = ( a + b [ u ] ) − [ f ( u ) u − µu ] ∈ L ∞ ( R , R ) , and s > > , we obtain u ∈ C ,γ ( R , R ) ∩ L ∞ ( R , R ) , for some γ > (see [51]) and that u > by themaximum principle. Now we prove that J µ ( u ) = c µ . Indeed, using u ∈ N µ , ( f ) and Fatou’s Lemma wehave c µ ≤ J µ ( u ) − h J ′ µ ( u ) , u i≤ lim inf n →∞ (cid:20) k u n k µ + Z R f ( u n ) u n − F ( u n ) dx (cid:21) = lim inf n →∞ J µ ( u n ) − h J ′ µ ( u n ) , u n i = c µ . Now, we consider the case u = 0 . Since c µ > and J µ is continuous, we can see that k u n k µ . FromLemma 3.3 it follows that we can define v n ( x ) = u n ( x + y n ) such that v n ⇀ v in H sµ for some v = 0 . Thenwe can argue as in the previous case to get the thesis. (cid:3) The next result shows an interesting relation between c ε and c V . Lemma 3.5. The numbers c ε and c V satisfy the following inequality lim sup ε → c ε ≤ c V . Proof. In the light of Lemma 3.4, we can find a positive ground state w ∈ H sV to (3.1), that is J ′ V ( w ) = 0 and J V ( w ) = c V . Since w ∈ C ,γ ( R , R ) ∩ L ∞ ( R , R ) , for some γ > , we get | w ( x ) | → as | x | → ∞ .Observing that w satisfies ( − ∆) s w + V a + bM w = ( a + b [ w ] ) − [ f ( w ) w − V w ] + V a + bM w in R , where < a ≤ a + b [ u ] ≤ a + bM , we can argue as in Lemma . in [24] to deduce the following decayestimate < w ( x ) ≤ C | x | s for | x | >> . (3.30)Now, let η ∈ C ∞ c ( R , [0 , be a cut-off function such that η = 1 in a neighborhood of zero B δ and supp( η ) ⊂ B δ ⊂ Λ for some δ > . Let us define w ε ( x ) := η ε ( x ) w ( x ) e ıA (0) · x , with η ε ( x ) = η ( ε x ) for ε > ,and we note that | w ε | = η ε w and w ε ∈ H sε in view of Lemma 2.3. Let us verify that lim ε → k w ε k ε = k w k V ∈ (0 , ∞ ) . (3.31)From the Dominated Convergence Theorem it follows that R R V ε ( x ) | w ε | dx → R R V | w | dx . Thus, it isonly need to prove that lim ε → [ w ε ] A ε = [ w ] . (3.32)By Lemma in [43], we know that [ η ε w ] → [ w ] as ε → . (3.33)On the other hand [ w ε ] A ε = Z Z R | e ıA (0) · x η ε ( x ) w ( x ) − e ıA ε ( x + y ) · ( x − y ) e ıA (0) · y η ε ( y ) w ( y ) | | x − y | s dxdy = [ η ε w ] + Z Z R η ε ( y ) w ( y ) | e ı [ A ε ( x + y ) − A (0)] · ( x − y ) − | | x − y | s dxdy + 2 ℜ Z Z R ( η ε ( x ) w ( x ) − η ε ( y ) w ( y )) η ε ( y ) w ( y )(1 − e − ı [ A ε ( x + y ) − A (0)] · ( x − y ) ) | x − y | s dxdy =: [ η ε w ] + X ε + 2 Y ε . In the light of | Y ε | ≤ [ η ε w ] √ X ε and (3.33), it is enough to see that X ε → as ε → to deduce that (3.32)holds. For all < β < α/ (1 + α − s ) , we get X ε ≤ Z R w ( y ) dy Z | x − y |≥ ε − β | e ı [ A ε ( x + y ) − A (0)] · ( x − y ) − | | x − y | s dx + Z R w ( y ) dy Z | x − y | <ε − β | e ı [ A ε ( x + y ) − A (0)] · ( x − y ) − | | x − y | s dx =: X ε + X ε . (3.34)Since | e ıt − | ≤ and w ∈ H s ( R , R ) , we have X ε ≤ C Z R w ( y ) dy Z ∞ ε − β ρ − − s dρ ≤ C ε βs → . (3.35) RACTIONAL KIRCHHOFF EQUATIONS WITH MAGNETIC FIELDS 17 Observing that | e ıt − | ≤ t for all t ∈ R , A ∈ C ,α ( R , R ) for α ∈ (0 , , and | x + y | ≤ | x − y | + 4 | y | ) ,we can deduce that X ε ≤ Z R w ( y ) dy Z | x − y | <ε − β | A ε (cid:0) x + y (cid:1) − A (0) | | x − y | s − dx ≤ C ε α Z R w ( y ) dy Z | x − y | <ε − β | x + y | α | x − y | s − dx ≤ C ε α Z R w ( y ) dy Z | x − y | <ε − β | x − y | s − − α dx + Z R | y | α w ( y ) dy Z | x − y | <ε − β | x − y | s − dx ! =: C ε α ( X , ε + X , ε ) . (3.36)Hence X , ε = C Z R w ( y ) dy Z ε − β ρ α − s dρ ≤ C ε − β (1+ α − s ) . (3.37)On the other hand, using (3.30), we have X , ε ≤ C Z R | y | α w ( y ) dy Z ε − β ρ − s dρ ≤ C ε − β (1 − s ) "Z B (0) w ( y ) dy + Z B c (0) | y | s ) − α dy ≤ C ε − β (1 − s ) . (3.38)From (3.34), (3.35), (3.36), (3.37) and (3.38) it follows that that X ε → , that is (3.31) holds true. Now,let t ε > be the unique number such that J ε ( t ε w ε ) = max t ≥ J ε ( tw ε ) . Clearly t ε satisfies t ε k w ε k ε + t ε [ w ε ] A ε = Z R g ( ε x, t ε | w ε | ) | t ε w ε | dx = Z R f ( t ε | w ε | ) | t ε w ε | dx, (3.39)where we used supp ( η ) ⊂ Λ and g ( t ) = f ( t ) on Λ .Now, we show that t ε → as ε → . Since η = 1 in B δ , w is a continuous positive function and f ( t ) t isincreasing for t > by ( f ) , we can deduce t ε k w ε k ε + b [ w ε ] A ε = Z R f ( t ε | w ε | ) t ε | w ε | | w ε | dx ≥ f ( t ε α ) t ε α Z B δ | w | dx where α := min ¯ B δ w > . Therefore, if t ε → ∞ as ε → , we can use (3.31) to see that b [ w ] = ∞ , anabsurd. On the other hand, if t ε → as ε → , by (3.39), the growth assumptions on g and (3.31) yield k w k V = 0 , which is impossible. Thus, t ε → t ∈ (0 , ∞ ) as ε → .Letting the limit as ε → in (3.39) and by (3.31), we can see that t k w k V + b [ w ] = Z R f ( t w )( t w ) w dx. By w ∈ N V and ( f ) , we can conclude that t = 1 . Applying the Dominated Convergence Theorem, wecan see that lim ε → J ε ( t ε w ε ) = J V ( w ) = c V . Using c ε ≤ max t ≥ J ε ( tw ε ) = J ε ( t ε w ε ) , we can infer that lim sup ε → c ε ≤ c V . (cid:3) Multiplicity result for the modified problem In this section we make use of the Ljusternik-Schnirelmann category theory to obtain multiple solutionsto (1.10). In particular, we relate the number of positive solutions of (1.10) to the topology of the set M .For this reason, we take δ > such that M δ = { x ∈ R : dist( x, M ) ≤ δ } ⊂ Λ , and we consider η ∈ C ∞ ( R + , [0 , such that η ( t ) = 1 if ≤ t ≤ δ and η ( t ) = 0 if t ≥ δ .For any y ∈ Λ , we introduce (see [7]) Ψ ε,y ( x ) = η ( | ε x − y | ) w (cid:18) ε x − yε (cid:19) e ıτ y ( ε x − yε ) , where τ y ( x ) = P j =1 A j ( x ) x j and w ∈ H s ( R ) is a positive ground state solution to the autonomousproblem (3.1) (see Lemma 3.4), and let t ε > be the unique number such that max t ≥ J ε ( t Ψ ε,y ) = J ε ( t ε Ψ ε,y ) . Finally, we consider Φ ε : M → N ε defined by setting Φ ε ( y ) = t ε Ψ ε,y . Lemma 4.1. The functional Φ ε satisfies the following limit lim ε → J ε (Φ ε ( y )) = c V uniformly in y ∈ M. Proof. Assume by contradiction that there exist δ > , ( y n ) ⊂ M and ε n → such that | J ε n (Φ ε n ( y n )) − c V | ≥ δ . (4.1)Using Lemma . in [7] and the Dominated Convergence Theorem we can observe that k Ψ ε n ,y n k ε n → k w k V ∈ (0 , ∞ ) . (4.2)On the other hand, since h J ′ ε n (Φ ε n ( y n )) , Φ ε n ( y n ) i = 0 and using the change of variable z = ε n x − y n ε n itfollows that t ε n k Ψ ε n ,y n k ε n + bt ε n [Ψ ε n ,y n ] A εn = Z R g ( ε n z + y n , | t ε n η ( | ε n z | ) w ( z ) | ) | t ε n η ( | ε n z | ) w ( z ) | dz. If z ∈ B δεn (0) ⊂ M δ ⊂ Λ , then ε n z + y n ∈ B δ ( y n ) ⊂ M δ ⊂ Λ ε . Since g ( x, t ) = f ( t ) for all x ∈ Λ and η ( t ) = 0 for t ≥ δ , we have t ε n k Ψ ε n ,y n k ε n + bt ε n [Ψ ε n ,y n ] A εn = Z R f ( | t ε n η ( | ε n z | ) w ( z ) | ) | t ε n η ( | ε n z | ) w ( z ) | . (4.3)In view of η = 1 in B δ (0) ⊂ B δεn (0) for all n large enough and (4.3) we can deduce that t ε n k Ψ ε n ,y n k ε n + b [Ψ ε n ,y n ] A εn = Z R f ( | t ε n Ψ ε n ,y n | ) | t ε n Ψ ε n ,y n | | Ψ ε n ,y n | dx ≥ Z B δ (0) f ( | t ε n η ( | ε n z | ) w ( z ) | ) | t ε n η ( | ε n z | ) w ( z ) | ( η ( | ε n z | ) w ( z )) dz = Z B δ (0) f ( | t ε n w ( z ) | ) | t ε n w ( z ) | w ( z ) dz ≥ f ( | t ε n w (ˆ z ) | ) | t ε n w (ˆ z ) | w (ˆ z ) | B δ (0) | , (4.4) RACTIONAL KIRCHHOFF EQUATIONS WITH MAGNETIC FIELDS 19 where w (ˆ z ) = min z ∈ B δ w ( z ) > . Now, assume by contradiction that t ε n → ∞ . This fact, (4.4) and (4.2) yield b [ w ] = ∞ , that is a contradiction. Hence, ( t ε n ) is bounded and, up to subsequence, we may assume that t ε n → t forsome t ≥ . In particular t > . In fact, if t = 0 , we can see that (3.26) and (4.3) imply that min { a, } r ≤ Z R f ( | t ε n η ( | ε n z | ) w ( z ) | ) | t ε n η ( | ε n z | ) w ( z ) | . In view of assumptions ( f ) - ( f ) and (4.2) we can deduce that t can not be zero. Hence t > . Thus,letting the limit as n → ∞ in (4.3), we can see that t k w k V + b [ w ] = Z R f (( t w ) )( t w ) w dx. Taking into account w ∈ N and using the fact that f ( t ) t is increasing by ( f ) , we can infer that t = 1 .Letting the limit as n → ∞ and using t ε n → we can conclude that lim n →∞ J ε n (Φ ε n ,y n ) = J V ( w ) = c V , which provides a contradiction in view of (4.1). (cid:3) For any δ > , we take ρ = ρ ( δ ) > such that M δ ⊂ B ρ , and we consider Υ : R → R defined by setting Υ ( x ) = (cid:26) x if | x | < ρ ρx | x | if | x | ≥ ρ. We define the barycenter map β ε : N ε → R as follows β ε ( u ) = Z R Υ ( ε x ) | u ( x ) | dx Z R | u ( x ) | dx . Arguing as Lemma . in [7], it is easy to see that the function β ε verifies the following limit: Lemma 4.2. lim ε → β ε (Φ ε ( y )) = y uniformly in y ∈ M. The next compactness result will play a fundamental role to prove that the solutions of (1.10) are alsosolution to (1.9). Lemma 4.3. Let ε n → and ( u n ) ⊂ N ε n be such that J ε n ( u n ) → c V . Then there exists (˜ y n ) ⊂ R such that v n ( x ) = | u n | ( x + ˜ y n ) has a convergent subsequence in H sV . Moreover, up to a subsequence, y n = ε n ˜ y n → y for some y ∈ M .Proof. Using h J ′ ε n ( u n ) , u n i = 0 , J ε n ( u n ) = c V + o n (1) , Lemma 3.5 and arguing as in the first part ofLemma 3.2, we can see that k u n k ε n ≤ C for all n ∈ N . Moreover, from Lemma 2.2, we also know that ( | u n | ) is bounded in H sV . Arguing as in the proof of Lemma 3.3, we can find a sequence (˜ y n ) ⊂ R , andconstants R > and β > such that lim inf n →∞ Z B R (˜ y n ) | u n | dx ≥ β > . (4.5)Put v n ( x ) = | u n | ( x + ˜ y n ) . Hence, ( v n ) is bounded in H sV and we may assume that v n ⇀ v in H s ( R , R ) as n → ∞ . Fix t n > such that ˜ v n = t n v n ∈ N V . Using Lemma 2.2, we can deduce that c V ≤ J V (˜ v n ) ≤ max t ≥ J ε n ( tv n ) = J ε n ( u n ) which together with Lemma 3.5 yields J V (˜ v n ) → c V . Moreover, ˜ v n in H sV . Since ( v n ) and (˜ v n ) arebounded in H sV and ˜ v n in H sV , we deduce that t n → t ∗ ≥ . Indeed t ∗ > since ˜ v n in H sV . From the uniqueness of the weak limit, we can deduce that ˜ v n ⇀ ˜ v = t ∗ v in H sV . This combined withLemma 3.4 implies that ˜ v n → ˜ v in H sV . (4.6)Consequently, v n → v in H sV as n → ∞ .Now, we set y n = ε n ˜ y n and we show that ( y n ) admits a subsequence, still denoted by y n , such that y n → y for some y ∈ M . We begin proving that ( y n ) is bounded. Assume by contradiction that, up toa subsequence, | y n | → ∞ as n → ∞ . Choose R > such that Λ ⊂ B R (0) . Then for n large enough, wehave | y n | > R , and for any z ∈ B R/ ε n it holds | ε n z + y n | ≥ | y n | − | ε n z | > R. Taking into account ( u n ) ⊂ N ε n , ( V ) , Lemma 2.2, the definition of g and the change of variable x z + ˜ y n ,we have a [ v n ] + Z R V v n dx ≤ a [ v n ] + Z R V v n dx + b [ v n ] ≤ Z R g ( ε n x + y n , | v n | ) | v n | dx ≤ Z B Rεn (0) ˜ f ( | v n | ) | v n | dx + Z R \ B Rεn (0) f ( | v n | ) | v n | + | v n | ∗ s dx ≤ V k Z R | v n | dx. which implies that v n → in H sV , that is a contradiction. Therefore, ( y n ) is bounded and we may assumethat y n → y ∈ R . If y / ∈ Λ , we can proceed as above to infer that v n → in H sV , which is impossible.Thus y ∈ Λ . Now, we aim to prove that V ( y ) = V . Assume by contradiction that V ( y ) > V . In thelight of (4.6), Fatou’s Lemma, the invariance of R by translations, Lemma 2.2 and Lemma 3.5, we obtain c V = J V (˜ v ) < a v ] + 12 Z R V ( y )˜ v dx + b v ] − Z R F ( | ˜ v | ) dx ≤ lim inf n →∞ h a v n ] + 12 Z R V ( ε n x + y n ) | ˜ v n | dx + b v n ] − Z R F ( | ˜ v n | ) dx i ≤ lim inf n →∞ h a t n | u n | ] + t n Z R V ( ε n z ) | u n | dz + b t n | u n | ] − Z R F ( | t n u n | ) dz i ≤ lim inf n →∞ J ε n ( t n u n ) ≤ lim inf n →∞ J ε n ( u n ) = c V which is an absurd. Therefore, in view of ( V ) , we can conclude that y ∈ M . (cid:3) Now, we consider the following subset of N ε e N ε = { u ∈ N ε : J ε ( u ) ≤ c V + h ( ε ) } , where h : R + → R + is such that h ( ε ) → as ε → . Fixed y ∈ M , we can use Lemma 4.1 to see that h ( ε ) = | J ε (Φ ε ( y )) − c V | → as ε → . Therefore Φ ε ( y ) ∈ e N ε , and e N ε = ∅ for any ε > . Arguing as inLemma . in [7], we have: Lemma 4.4. For any δ > , there holds that lim ε → sup u ∈ e N ε dist( β ε ( u ) , M δ ) = 0 . We end this section proving a multiplicity result for (1.10). Theorem 4.1. For any δ > such that M δ ⊂ Λ , there exists ˜ ε δ > such that, for any ε ∈ (0 , ε δ ) , problem (1.10) has at least cat M δ ( M ) nontrivial solutions.Proof. Given δ > such that M δ ⊂ Λ , we can use Lemma 4.2, Lemma 4.1, Lemma 4.4 and argue as in [14]to deduce the existence of ˜ ε δ > such that, for any ε ∈ (0 , ε δ ) , the following diagram M Φ ε → e N ε β ε → M δ RACTIONAL KIRCHHOFF EQUATIONS WITH MAGNETIC FIELDS 21 is well defined and β ε ◦ Φ ε is homotopically equivalent to the embedding ι : M → M δ . Hence, cat e N ε ( e N ε ) ≥ cat M δ ( M ) . From Proposition 3.1 and standard Ljusternik-Schnirelmann theory, we can deduce that J ε possesses at least cat e N ε ( e N ε ) critical points on N ε . In view of Corollary 3.1, we obtain cat M δ ( M ) nontrivialsolutions for (1.10). (cid:3) Proof of Theorem 1.1 This last section is devoted to the proof of the main result of this paper. In order to show that thesolutions of (1.10) are indeed solutions to (1.9), we need to verify that (1.11) holds true. For this purpose,we begin proving the following fundamental result in which we use a variant of Moser iteration scheme [41]and a Kato’s approximation argument [34]. Lemma 5.1. Let ε n → and u n ∈ e N ε n be a solution to (1.10) . Then v n = | u n | ( · + ˜ y n ) satisfies v n ∈ L ∞ ( R , R ) and there exists C > such that k v n k L ∞ ( R ) ≤ C for all n ∈ N , where ˜ y n is given by Lemma 4.3. Moreover lim | x |→∞ v n ( x ) = 0 uniformly in n ∈ N . Proof. For any L > we define u L,n := min {| u n | , L } ≥ and we set v L,n = u β − L,n u n , where β > willbe chosen later. Taking v L,n as test function in (1.10) we can see that ( a + b [ u n ] A εn ) ℜ Z Z R ( u n ( x ) − u n ( y ) e ıA ( x + y ) · ( x − y ) ) | x − y | s ( u n u β − L,n ( x ) − u n u β − L,n ( y ) e ıA ( x + y ) · ( x − y ) ) dxdy ! = Z R g ( ε n x, | u n | ) | u n | u β − L,n dx − Z R V ( ε n x ) | u n | u β − L,n dx. (5.1)Now, we observe that ℜ (cid:20) ( u n ( x ) − u n ( y ) e ıA ( x + y ) · ( x − y ) )( u n u β − L,n ( x ) − u n u β − L,n ( y ) e ıA ( x + y ) · ( x − y ) ) (cid:21) = ℜ h | u n ( x ) | v β − L ( x ) − u n ( x ) u n ( y ) u β − L,n ( y ) e − ıA ( x + y ) · ( x − y ) − u n ( y ) u n ( x ) u β − L,n ( x ) e ıA ( x + y ) · ( x − y ) + | u n ( y ) | u β − L,n ( y ) i ≥ ( | u n ( x ) | u β − L,n ( x ) − | u n ( x ) || u n ( y ) | u β − L,n ( y ) − | u n ( y ) || u n ( x ) | u β − L,n ( x ) + | u n ( y ) | u β − L,n ( y )= ( | u n ( x ) | − | u n ( y ) | )( | u n ( x ) | u β − L,n ( x ) − | u n ( y ) | u β − L,n ( y )) . Thus ℜ Z Z R ( u n ( x ) − u n ( y ) e ıA ( x + y ) · ( x − y ) ) | x − y | s ( u n u β − L,n ( x ) − u n u β − L,n ( y ) e ıA ( x + y ) · ( x − y ) ) dxdy ! ≥ Z Z R ( | u n ( x ) | − | u n ( y ) | ) | x − y | s ( | u n ( x ) | u β − L,n ( x ) − | u n ( y ) | u β − L,n ( y )) dxdy. (5.2)For all t ≥ , we define γ ( t ) = γ L,β ( t ) = tt β − L , where t L = min { t, L } . Since γ is an increasing function, we have ( p − q )( γ ( p ) − γ ( q )) ≥ for any p, q ∈ R . Let us consider the functions Λ( t ) = | t | and Γ( t ) = Z t ( γ ′ ( τ )) dτ. and we note that Λ ′ ( p − q )( γ ( p ) − γ ( q )) ≥ | Γ( p ) − Γ( q ) | for any p, q ∈ R . (5.3) Indeed, for any p, q ∈ R such that p < q , the Jensen inequality yields Λ ′ ( p − q )( γ ( p ) − γ ( q )) = ( p − q ) Z pq γ ′ ( t ) dt = ( p − q ) Z pq (Γ ′ ( t )) dt ≥ (cid:18)Z pq Γ ′ ( t ) dt (cid:19) = (Γ( p ) − Γ( q )) . In a similar way, we can prove that if p ≥ q then Λ ′ ( p − q )( γ ( p ) − γ ( q )) ≥ (Γ( q ) − Γ( p )) , that is (5.3)holds. Hence, in view of (5.3), we can deduce that | Γ( | u n ( x ) | ) − Γ( | u n ( y ) | ) | ≤ ( | u n ( x ) | − | u n ( y ) | )(( | u n | u β − L,n )( x ) − ( | u n | u β − L,n )( y )) . (5.4)By (5.2) and (5.4), it follows that ℜ Z Z R ( u n ( x ) − u n ( y ) e ıA ( x + y ) · ( x − y ) ) | x − y | s ( u n u β − L,n ( x ) − u n u β − L,n ( y ) e ıA ( x + y ) · ( x − y ) ) dxdy ! ≥ [Γ( | u n | )] . (5.5)Observing that Γ( | u n | ) ≥ β | u n | u β − L,n and recalling the fractional Sobolev embedding D s, ( R , R ) ⊂ L ∗ s ( R , R ) (see [19]), we get [Γ( | u n | )] ≥ S ∗ k Γ( | u n | ) k L ∗ s ( R ) ≥ (cid:18) β (cid:19) S ∗ k| u n | u β − L,n k L ∗ s ( R ) . (5.6)Putting together (5.1), (5.5), (5.6) and noting that a ≤ a + b [ u n ] A εn ≤ a + bM , we obtain that a (cid:18) β (cid:19) S ∗ k| u n | u β − L,n k L ∗ s ( R ) + Z R V ( ε n x ) | u n | u β − L,n dx ≤ Z R g ( ε n x, | u n | ) | u n | u β − L,n dx. (5.7)Now, by ( g ) and ( g ) , it follows that for any ξ > there exists C ξ > such that g ( ε n x, t ) t ≤ ξ | t | + C ξ | t | ∗ s for all t ∈ R . (5.8)Taking ξ ∈ (0 , V ) and using (5.7) and (5.8) we have k w L,n k L ∗ s ( R ) ≤ Cβ Z R | u n | ∗ s u β − L,n , (5.9)where we set w L,n := | u n | u β − L,n .Take β = ∗ s and fix R > . Recalling that ≤ u L,n ≤ | u n | and applying Hölder inequality, we get Z R | u n | ∗ s u β − L,n dx = Z R | u n | ∗ s − | u n | u ∗ s − L,n dx = Z R | u n | ∗ s − ( | u n | u ∗ s − L,n ) dx ≤ Z {| u n | Since ( | u n | ) is bounded in H s ( R , R ) , we can see that for any R sufficiently large Z {| u n | >R } | u n | ∗ s dx ! ∗ s − ∗ s ≤ β . (5.11)In the light of (5.9), (5.10) and (5.11), we infer that (cid:18)Z R ( | u n | u ∗ s − L,n ) ∗ s (cid:19) ∗ s ≤ Cβ Z R R ∗ s − | u n | ∗ s dx < ∞ , and taking the limit as L → ∞ we deduce that | u n | ∈ L (2 ∗ s )22 ( R , R ) .Using ≤ u L,n ≤ | u n | and passing to the limit as L → ∞ in (5.9), we have k u n k βL β ∗ s ( R ) ≤ Cβ Z R | u n | ∗ s +2( β − , from which we deduce that (cid:18)Z R | u n | β ∗ s dx (cid:19) β − ∗ s ≤ Cβ β − (cid:18)Z R | u n | ∗ s +2( β − (cid:19) β − . For m ≥ we define β m +1 inductively so that ∗ s + 2( β m +1 − 1) = 2 ∗ s β m and β = ∗ s .Therefore (cid:18)Z R | u n | β m +1 ∗ s dx (cid:19) βm +1 − ∗ s ≤ Cβ βm +1 − m +1 (cid:18)Z R | u n | ∗ s β m (cid:19) ∗ s ( βm − . Let us define D m = (cid:18)Z R | u n | ∗ s β m (cid:19) ∗ s ( βm − . Using an iteration argument, we can find C > independent of m such that D m +1 ≤ m Y k =1 Cβ βk +1 − k +1 D ≤ C D . Taking the limit as m → ∞ we get k u n k L ∞ ( R ) ≤ C D =: K for all n ∈ N . (5.12)Moreover, by interpolation, we can deduce that ( | u n | ) strongly converges in L r ( R , R ) for all r ∈ (2 , ∞ ) .From the growth assumptions on g , we can see that g ( ε x, | u n | ) | u n | strongly converges in L r ( R , R ) forall r ∈ [2 , ∞ ) .In what follows, we prove that | u n | is a weak subsolution to (cid:26) ( a + b [ v ] )( − ∆) s v + V v = g ( ε n x, v ) v in R v ≥ in R . (5.13)Roughly speaking, we will prove a Kato’s inequality for the modulus of solutions of (1.10).Fix ϕ ∈ C ∞ c ( R , R ) such that ϕ ≥ , and we take ψ δ,n = u n u δ,n ϕ as test function in (1.9), where u δ,n = p | u n | + δ for δ > . Note that ψ δ,n ∈ H sε n for all δ > and n ∈ N . Indeed R R V ( ε n x ) | ψ δ,n | dx ≤ R supp( ϕ ) V ( ε n x ) ϕ dx < ∞ . Now, we can see that ψ δ,n ( x ) − ψ δ,n ( y ) e ıA ε ( x + y ) · ( x − y ) = (cid:18) u n ( x ) u δ,n ( x ) (cid:19) ϕ ( x ) − (cid:18) u n ( y ) u δ,n ( y ) (cid:19) ϕ ( y ) e ıA ε ( x + y ) · ( x − y ) = (cid:20)(cid:18) u n ( x ) u δ,n ( x ) (cid:19) − (cid:18) u n ( y ) u δ,n ( x ) (cid:19) e ıA ε ( x + y ) · ( x − y ) (cid:21) ϕ ( x )+ [ ϕ ( x ) − ϕ ( y )] (cid:18) u n ( y ) u δ,n ( x ) (cid:19) e ıA ε ( x + y ) · ( x − y ) + (cid:18) u n ( y ) u δ,n ( x ) − u n ( y ) u δ,n ( y ) (cid:19) ϕ ( y ) e ıA ε ( x + y ) · ( x − y ) , and using | z + w + k | ≤ | z | + | w | + | k | ) for all z, w, k ∈ C , | e ıt | = 1 for all t ∈ R , u δ,n ≥ δ , | u n u δ,n | ≤ ,(5.12) and | p | z | + δ − p | w | + δ | ≤ || z | − | w || for all z, w ∈ C , we can deduce that | ψ δ,n ( x ) − ψ δ,n ( y ) e ıA ε ( x + y ) · ( x − y ) | ≤ δ | u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) | k ϕ k L ∞ ( R ) + 4 δ | ϕ ( x ) − ϕ ( y ) | k| u n |k L ∞ ( R ) + 4 δ k| u n |k L ∞ ( R ) k ϕ k L ∞ ( R ) | u δ,n ( y ) − u δ,n ( x ) | ≤ δ | u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) | k ϕ k L ∞ ( R ) + 4 K δ | ϕ ( x ) − ϕ ( y ) | + 4 K δ k ϕ k L ∞ ( R ) || u n ( y ) | − | u n ( x ) || . In view of u n ∈ H sε n , | u n | ∈ H s ( R , R ) (by Lemma 2.2) and ϕ ∈ C ∞ c ( R , R ) , we get ψ δ,n ∈ H sε n .Therefore ( a + b [ u n ] A εn ) ℜ "Z Z R ( u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) ) | x − y | s u n ( x ) u δ,n ( x ) ϕ ( x ) − u n ( y ) u δ,n ( y ) ϕ ( y ) e − ıA ε ( x + y ) · ( x − y ) ! dxdy + Z R V ( ε x ) | u n | u δ,n ϕdx = Z R g ( ε x, | u n | ) | u n | u δ,n ϕdx. (5.14)From ℜ ( z ) ≤ | z | for all z ∈ C and | e ıt | = 1 for all t ∈ R , it follows that ℜ " ( u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) ) u n ( x ) u δ,n ( x ) ϕ ( x ) − u n ( y ) u δ,n ( y ) ϕ ( y ) e − ıA ε ( x + y ) · ( x − y ) ! = ℜ " | u n ( x ) | u δ,n ( x ) ϕ ( x ) + | u n ( y ) | u δ,n ( y ) ϕ ( y ) − u n ( x ) u n ( y ) u δ,n ( y ) ϕ ( y ) e − ıA ε ( x + y ) · ( x − y ) − u n ( y ) u n ( x ) u δ,n ( x ) ϕ ( x ) e ıA ε ( x + y ) · ( x − y ) ≥ (cid:20) | u n ( x ) | u δ,n ( x ) ϕ ( x ) + | u n ( y ) | u δ,n ( y ) ϕ ( y ) − | u n ( x ) | | u n ( y ) | u δ,n ( y ) ϕ ( y ) − | u n ( y ) | | u n ( x ) | u δ,n ( x ) ϕ ( x ) (cid:21) . (5.15)Now, we can note that | u n ( x ) | u δ,n ( x ) ϕ ( x ) + | u n ( y ) | u δ,n ( y ) ϕ ( y ) − | u n ( x ) | | u n ( y ) | u δ,n ( y ) ϕ ( y ) − | u n ( y ) | | u n ( x ) | u δ,n ( x ) ϕ ( x )= | u n ( x ) | u δ,n ( x ) ( | u n ( x ) | − | u n ( y ) | ) ϕ ( x ) − | u n ( y ) | u δ,n ( y ) ( | u n ( x ) | − | u n ( y ) | ) ϕ ( y )= (cid:20) | u n ( x ) | u δ,n ( x ) ( | u n ( x ) | − | u n ( y ) | ) ϕ ( x ) − | u n ( x ) | u δ,n ( x ) ( | u n ( x ) | − | u n ( y ) | ) ϕ ( y ) (cid:21) + (cid:18) | u n ( x ) | u δ,n ( x ) − | u n ( y ) | u δ,n ( y ) (cid:19) ( | u n ( x ) | − | u n ( y ) | ) ϕ ( y )= | u n ( x ) | u δ,n ( x ) ( | u n ( x ) | − | u n ( y ) | )( ϕ ( x ) − ϕ ( y )) + (cid:18) | u n ( x ) | u δ,n ( x ) − | u n ( y ) | u δ,n ( y ) (cid:19) ( | u n ( x ) | − | u n ( y ) | ) ϕ ( y ) ≥ | u n ( x ) | u δ,n ( x ) ( | u n ( x ) | − | u n ( y ) | )( ϕ ( x ) − ϕ ( y )) (5.16)where in the last inequality we used that (cid:18) | u n ( x ) | u δ,n ( x ) − | u n ( y ) | u δ,n ( y ) (cid:19) ( | u n ( x ) | − | u n ( y ) | ) ϕ ( y ) ≥ due to h ( t ) = t √ t + δ is increasing for t ≥ and ϕ ≥ in R . RACTIONAL KIRCHHOFF EQUATIONS WITH MAGNETIC FIELDS 25 Observing that | | u n ( x ) | u δ,n ( x ) ( | u n ( x ) | − | u n ( y ) | )( ϕ ( x ) − ϕ ( y )) || x − y | s ≤ || u n ( x ) | − | u n ( y ) ||| x − y | s | ϕ ( x ) − ϕ ( y ) || x − y | s ∈ L ( R ) , and | u n ( x ) | u δ,n ( x ) → a.e. in R as δ → , we can apply (5.15), (5.16) and the Dominated Convergence Theoremto infer that lim sup δ → ℜ "Z Z R ( u n ( x ) − u n ( y ) e ıA ε ( x + y ) · ( x − y ) ) | x − y | s u n ( x ) u δ,n ( x ) ϕ ( x ) − u n ( y ) u δ,n ( y ) ϕ ( y ) e − ıA ε ( x + y ) · ( x − y ) ! dxdy ≥ lim sup δ → Z Z R | u n ( x ) | u δ,n ( x ) ( | u n ( x ) | − | u n ( y ) | )( ϕ ( x ) − ϕ ( y )) dxdy | x − y | s = Z Z R ( | u n ( x ) | − | u n ( y ) | )( ϕ ( x ) − ϕ ( y )) | x − y | s dxdy. (5.17)On the other hand, from the Dominated Convergence Theorem again (we recall that | u n | u δ,n ≤ | u n | and ϕ ∈ C ∞ c ( R , R ) ) we can see that lim δ → Z R V ( ε n x ) | u n | u δ,n ϕdx = Z R V ( ε n x ) | u n | ϕdx ≥ Z R V | u n | ϕdx (5.18)and lim δ → Z R g ( ε n x, | u n | ) | u n | u δ,n ϕdx = Z R g ( ε n x, | u n | ) | u n | ϕdx. (5.19)By Lemma 2.2 we can also see that lim sup n →∞ ( a + b [ u n ] A εn ) ≥ ( a + b [ | u n | ] ) . (5.20)Putting together (5.14), (5.17), (5.18), (5.19) and (5.20) we can deduce that ( a + b [ | u n | ] ) Z Z R ( | u n ( x ) | − | u n ( y ) | )( ϕ ( x ) − ϕ ( y )) | x − y | s dxdy + Z R V | u n | ϕdx ≤ Z R g ( ε n x, | u n | ) | u n | ϕdx for any ϕ ∈ C ∞ c ( R , R ) such that ϕ ≥ , that is | u n | is a weak subsolution to (5.13).Now, we set v n = | u n | ( · + ˜ y n ) . Then Lemma 2.2 yields a + b [ v n ] = a + b [ | u n | ] ≤ a + b [ u n ] A εn ≤ a + bM . We also note that v n satisfies ( − ∆) s v n + V a + bM v n ≤ g n in R , (5.21)where g n := ( a + b [ v n ] ) − [ g ( ε n x + ε n ˜ y n , v n ) v n − V v n ] + V a + bM v n . Let z n ∈ H s ( R , R ) be the unique solution to ( − ∆) s z n + V a + bM z n = g n in R . (5.22)In the light of (5.12), we know that k v n k L ∞ ( R ) ≤ C for all n ∈ N , and by interpolation v n → v stronglyconverges in L r ( R , R ) for all r ∈ (2 , ∞ ) , for some v ∈ L r ( R , R ) . From the growth assumptions on g , wecan see that g n → ( a + b [ v ] ) − [ f ( v ) v − V v ] + V a + bM v in L r ( R , R ) ∀ r ∈ [2 , ∞ ) , and there exists C > such that k g n k L ∞ ( R ) ≤ C for all n ∈ N . Then z n = K ∗ g n (see [24]), where K isthe Bessel kernel, and arguing as in [2], we can prove that | z n ( x ) | → as | x | → ∞ uniformly with respectto n ∈ N . Taking into account v n satisfies (5.31) and z n solves (5.22), we can use a comparison argumentto see that ≤ v n ≤ z n a.e. in R and for all n ∈ N . In conclusion, v n ( x ) → as | x | → ∞ uniformly withrespect to n ∈ N . (cid:3) Now, we are ready to give the proof of Theorem 1.1. Proof of Theorem 1.1. Let δ > be such that M δ ⊂ Λ , and we show that there exists ˆ ε δ > such that forany ε ∈ (0 , ˆ ε δ ) and any solution u ∈ e N ε of (1.10), it holds k u k L ∞ ( R \ Λ ε ) < t a ′ . (5.23)We argue by contradiction, and assume that there is a sequence ε n → , u n ∈ e N ε n such that k u n k L ∞ ( R \ Λ ε ) ≥ t a ′ . (5.24)Since J ε n ( u n ) ≤ c V + h ( ε n ) , we can argue as in the first part of Lemma 4.3 to deduce that J ε n ( u n ) → c V .In view of Lemma 4.3, there exists (˜ y n ) ⊂ R such that ε n ˜ y n → y for some y ∈ M . Take r > suchthat, for some subsequence still denoted by itself, it holds B r (˜ y n ) ⊂ Λ for all n ∈ N . Hence B rεn (˜ y n ) ⊂ Λ ε n n ∈ N . Consequently, R \ Λ ε n ⊂ R \ B rεn (˜ y n ) for any n ∈ N . By Lemma 5.1, we can find R > such that v n ( x ) < t a ′ for | x | ≥ R, n ∈ N , where v n ( x ) = | u ε n | ( x + ˜ y n ) ( v n is also strongly convergent in H s ( R , R ) ), from which we deduce that | u ε n ( x ) | < a for any x ∈ R \ B R (˜ y n ) and n ∈ N . Then, there exists ν ∈ N such that for any n ≥ ν and r/ ε n > R it holds R \ Λ ε n ⊂ R \ B rεn (˜ y n ) ⊂ R \ B R (˜ y n ) . Therefore, | u ε n ( x ) | < a for any x ∈ R \ Λ ε n and n ≥ ν , and this is impossible by (5.24).Let ˜ ε δ > be given by Theorem 4.1 and we set ε δ = min { ˜ ε δ , ˆ ε δ } . Applying Theorem 4.1 we obtain cat M δ ( M ) nontrivial solutions to (1.10). If u ∈ H sε is one of these solutions, then u ∈ e N ε , and in view of(5.23) and the definition of g we can infer that u is also a solution to (1.10). Since ˆ u ε ( x ) = u ε ( x/ ε ) is asolution to (1.1), we can infer that (1.1) has at least cat M δ ( M ) nontrivial solutions.Finally, we investigate the behavior of the maximum points of | ˆ u ε n | . Take ε n → and ( u ε n ) a sequenceof solutions to (1.10) as above. From ( g ) , we can find γ > such that g ( ε x, t ) t ≤ V t , for all x ∈ R , | t | ≤ γ. (5.25)Arguing as above, we can find R > such that k u ε n k L ∞ ( B cR (˜ y n )) < γ. (5.26)Up to a subsequence, we may also assume that k u ε n k L ∞ ( B R (˜ y n )) ≥ γ. (5.27)Indeed, if (5.27) does not hold, we get k u ε n k L ∞ ( R ) < γ , and using J ′ ε n ( u ε n ) = 0 , (5.25) and Lemma 2.2we can deduce that a [ | u ε n | ] + Z R V | u ε n | dx ≤ k u ε n k ε n + b [ u ε n ] A εn = Z R g ε n ( x, | u ε n | ) | u ε n | dx ≤ V Z R | u ε n | dx. This fact yields k u ε n k H s ( R ) = 0 , which is impossible. Hence, (5.27) is verified.In the light of (5.26) and (5.27), we can see that the maximum points p n of | u ε n | belong to B R (˜ y n ) , thatis p n = ˜ y n + q n for some q n ∈ B R . Since the associated solution of (1.1) is of the form ˆ u n ( x ) = u ε n ( x/ ε n ) ,we can infer that a maximum point η ε n of | ˆ u n | is η ε n = ε n ˜ y n + ε n q n . Since q n ∈ B R , ε n ˜ y n → y and V ( y ) = V , we can use the continuity of V to deduce that lim n →∞ V ( η ε n ) = V . RACTIONAL KIRCHHOFF EQUATIONS WITH MAGNETIC FIELDS 27 Finally, we provide a decay estimate for | ˆ u n | . Using Lemma . in [24], there exists a function w such that < w ( x ) ≤ C | x | s , (5.28)and ( − ∆) s w + V a + bM ) w ≥ in R \ B R (5.29)for some suitable R > , and M > is such that a + bM ≥ a + b [ u n ] A εn ≥ a + b [ v n ] (the last inequalityis due to Lemma 2.2). By Lemma 5.1, we know that v n ( x ) → as | x | → ∞ uniformly in n ∈ N , so we canuse ( g ) to deduce that there exists R > such that g ( ε n x + ε n ˜ y n , v n ) v n ≤ V v n in B cR . (5.30)Arguing as in Lemma 5.1, we can note that v n verifies ( − ∆) s v n + V a + bM v n ≤ g n in R , (5.31)where g n := ( a + b [ v n ] ) − [ g ( ε n x + ε n ˜ y n , v n ) v n − V ( ε n x + ε n ˜ y n ) v n ] + V a + bM v n . Let us denote by w n the unique solution to ( − ∆) s w n + V ( a + bM ) w n = g n in R . By comparison, we have ≤ v n ≤ w n in R and together with (5.30) we get ( − ∆) s w n + V a + bM ) w n = ( − ∆) s w n + V ( a + bM ) w n − V a + bM ) w n ≤ g n − V a + bM ) v n ≤ ( a + b [ v n ] ) − [ g ( ε n x + ε n ˜ y n , v n ) v n − V ( ε n x + ε n ˜ y n ) v n ] + V a + bM ) v n ≤ ( a + b [ v n ] ) − (cid:26) g ( ε n x + ε n ˜ y n , v n ) v n − (cid:18) V ( ε n x + ε n ˜ y n ) − V (cid:19) v n (cid:27) ≤ ( a + b [ v n ] ) − (cid:26) g ( ε n x + ε n ˜ y n , v n ) v n − V v n (cid:27) ≤ in B cR . Choose R = max { R , R } and we set c = inf B R w > and ˜ w n = ( d + 1) w − cw n . (5.32)where d = sup n ∈ N k w n k L ∞ ( R ) < ∞ . Our purpose is to verify that ˜ w n ≥ in R . (5.33)Let us note that lim | x |→∞ sup n ∈ N ˜ w n ( x ) = 0 , (5.34) ˜ w n ≥ dc + w − dc > in B R , (5.35) ( − ∆) s ˜ w n + V a + bM ) ˜ w n ≥ in R \ B R . (5.36)Now, we argue by contradiction. Suppose that there exists a sequence (¯ x j,n ) ⊂ R such that inf x ∈ R ˜ w n ( x ) = lim j →∞ ˜ w n (¯ x j,n ) < . (5.37) From (5.34), it follows that (¯ x j,n ) is bounded, and, up to subsequence, we may assume that there exists ¯ x n ∈ R such that ¯ x j,n → ¯ x n as j → ∞ . Thus, (5.37) gives inf x ∈ R ˜ w n ( x ) = ˜ w n (¯ x n ) < . (5.38)Using the minimality of ¯ x n and the representation formula for the fractional Laplacian (see Lemma . in [19]), we can deduce that ( − ∆) s ˜ w n (¯ x n ) = c ,s Z R w n (¯ x n ) − ˜ w n (¯ x n + ξ ) − ˜ w n (¯ x n − ξ ) | ξ | s dξ ≤ . (5.39)In view of (5.35) and (5.37), we get ¯ x n ∈ R \ B R , which together with (5.38) and (5.39) yields ( − ∆) s ˜ w n (¯ x n ) + V a + bM ) ˜ w n (¯ x n ) < , which is a contradiction due to (5.36). 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