Concentrating solutions for a fractional Kirchhoff equation with critical growth
aa r X i v : . [ m a t h . A P ] J un CONCENTRATING SOLUTIONS FOR A FRACTIONAL KIRCHHOFF EQUATIONWITH CRITICAL GROWTH
VINCENZO AMBROSIO
Abstract.
In this paper we consider the following class of fractional Kirchhoff equations with critical growth: ( (cid:16) ε s a + ε s − b R R | ( − ∆) s u | dx (cid:17) ( − ∆) s u + V ( x ) u = f ( u ) + | u | ∗ s − u in R ,u ∈ H s ( R ) , u > in R , where ε > is a small parameter, a, b > are constants, s ∈ ( , , ∗ s = − s is the fractional criticalexponent, ( − ∆) s is the fractional Laplacian operator, V is a positive continuous potential and f is a superlinearcontinuous function with subcritical growth. Using penalization techniques and variational methods, we provethe existence of a family of positive solutions u ε which concentrates around a local minimum of V as ε → . Introduction
This paper is devoted to the existence and concentration of positive solutions for the following fractionalKirchhoff type equation with critical nonlinearity: ( (cid:16) ε s a + ε s − b R R | ( − ∆) s u | dx (cid:17) ( − ∆) s u + V ( x ) u = f ( u ) + | u | ∗ s − u in R ,u ∈ H s ( R ) , u > in R , (1.1)where ε > is a small parameter, a, b > are constants, s ∈ ( , is fixed, ∗ s = − s is the fractionalcritical exponent, and ( − ∆) s is the fractional Laplacian operator, which (up to normalization factors) maybe defined for smooth functions u : R → R as ( − ∆) s u ( x ) = − Z R u ( x + y ) + u ( x − y ) − u ( x ) | y | s dy ( x ∈ R ) , (see [18, 36] and the references therein for further details and applications).The potential V : R → R is a continuous function satisfying the following conditions introduced by del Pinoand Felmer in [17]: ( V ) V := inf x ∈ R V ( x ) > , ( V ) there exists a bounded open set Λ ⊂ R such that < V := inf Λ V < min ∂ Λ V, while f : R → R is a continuous function fulfilling the following hypotheses: ( f ) f ( t ) = o ( t ) as t → + , ( f ) there exist q, σ ∈ (4 , ∗ s ) , C > such that f ( t ) ≥ C t q − ∀ t > , lim t →∞ f ( t ) t σ − = 0 , ( f ) there exists ϑ ∈ (4 , ∗ s ) such that < ϑF ( t ) ≤ tf ( t ) for all t > , ( f ) the map t f ( t ) t is increasing in (0 , ∞ ) .Since we will look for positive solutions to (1.1), we assume that f ( t ) = 0 for t ≤ .We note that when a = 1 , b = 0 and R is replaced by R N , then (1.1) reduces to a fractional Schrödingerequation of the type ε s ( − ∆) s u + V ( x ) u = h ( x, u ) in R N , (1.2)which has been introduced by Laskin [32] as a result of expanding the Feynman path integral, from theBrownian like to the Lévy like quantum mechanical paths. Equation (1.2) has received a great interest bymany mathematicians, and several results have been obtained under different and suitable assumptions on V and h ; see for instance [4, 6–8, 16, 19–21, 30, 44, 46] and the references therein. In particular way, the existence Mathematics Subject Classification.
Key words and phrases. fractional Kirchhoff equation; variational methods; critical growth. and concentration as ε → of positive solutions to (1.2) has been widely investigated in recent years. Forinstance, Dávila et al. [16] showed via Lyapunov-Schmidt reduction, that if the potential V satisfies V ∈ C ,α ( R N ) ∩ L ∞ ( R N ) and inf x ∈ R N V ( x ) > , then (1.1) has multi-peak solutions. Shang et al. [46] used Ljusternik-Schnirelmann theory to obtain multiplepositive solutions for a fractional Schrödinger equation with critical growth assuming that the potential V : R N → R fulfills the following assumption proposed by Rabinowitz [43]: V ∞ := lim inf | x |→∞ V ( x ) > inf x ∈ R N V ( x ) =: V , where V ∞ ∈ (0 , ∞ ] . (V)Fall et al. [20] established necessary and sufficient conditions on the smooth potential V in order to produceconcentration of solutions of (1.1) when the parameter ε converges to zero. Moreover, when V is coerciveand has a unique global minimum, then ground-states concentrate at this point. Alves and Miyagaki [4] (seealso [7]) studied the existence and concentration of positive solutions to (1.1), via a penalization approach,under assumptions ( V ) - ( V ) and f is a subcritical nonlinearity.On the other hand, if we set s = ε = 1 and we replace f ( u ) + | u | ∗ s − u by a more general nonlinearity h ( x, u ) , then (1.1) becomes the well-known classical Kirchhoff equation − (cid:18) a + b Z R |∇ u | dx (cid:19) ∆ u + V ( x ) u = h ( x, u ) in R , (1.3)which is related to the stationary analogue of the Kirchhoff equation ρu tt − (cid:18) p h + E L Z L | u x | dx (cid:19) u xx = 0 , (1.4)introduced by Kirchhoff [31] in as an extension of the classical D’Alembert’s wave equation for describingthe transversal oscillations of a stretched string. Here L is the length of the string, h is the area of the cross-section, E is the young modulus (elastic modulus) of the material, ρ is the mass density, and p is the initialtension. We refer to [12, 40] for the early classical studies dedicated to (1.4). We also note that nonlocalboundary value problems like (1.3) model several physical and biological systems where u describes a processwhich depends on the average of itself, as for example, the population density; see [2, 14]. However, onlyafter the Lions’ work [33], where a functional analysis approach was proposed to attack a general Kirchhoffequation in arbitrary dimension with external force term, problem (1.3) began to catch the attention ofseveral mathematicians; see [1, 13, 24, 27, 28, 48] and the references therein. For instance, He and Zou [28]obtained existence and multiplicity results for small ε > of the following perturbed Kirchhoff equation − (cid:18) aε + bε Z R |∇ u | dx (cid:19) ∆ u + V ( x ) u = g ( u ) in R , (1.5)where the potential V satisfies condition ( V ) and g is a subcritical nonlinearity. Wang et al. [48] studied themultiplicity and concentration phenomenon for (1.5) when g ( u ) = λf ( u ) + | u | u , f is a continuous subcriticalnonlinearity and λ is large. Figueiredo and Santos Junior [24] used the generalized Nehari manifold methodto obtain a multiplicity result for a subcritical Kirchhoff equation under conditions ( V ) - ( V ) . He et al. [27]dealt with the existence and multiplicity of solutions to (1.5), where g ( u ) = f ( u ) + u , f ∈ C is a subcriticalnonlinearity which does not satisfies the Ambrosetti-Rabinowitz condition [5] and V fulfills ( V ) - ( V ) .In the nonlocal framework, Fiscella and Valdinoci [26] proposed for the first time a stationary fractionalKirchhoff variational model in a bounded domain Ω ⊂ R N with homogeneous Dirichlet boundary conditionsand involving a critical nonlinearity: ( M (cid:16)R R N | ( − ∆) s u | dx (cid:17) ( − ∆) s u = λf ( x, u ) + | u | ∗ s − u in Ω ,u = 0 in R N \ Ω , (1.6)where M is a continuous Kirchhoff function whose model case is given by M ( t ) = a + bt . Their model takes careof the nonlocal aspect of the tension arising from nonlocal measurements of the fractional length of the string;see [26] for more details. After the pioneering work [26], several authors dealt with existence and multiplicityof solutions for (1.6); see [11, 23, 36, 38] and their references. On the other hand, some interesting results forfractional Kirchhoff equations in R N have been established in [9, 10, 25, 34, 35, 41, 42]. For instance, Pucciand Saldi [41] obtained the existence and multiplicity of nontrivial solutions for a Kirchhoff type eigenvalueproblem in R N involving a critical nonlinearity. Fiscella and Pucci [25] dealt with stationary fractional RACTIONAL CRITICAL KIRCHHOFF EQUATIONS 3
Kirchhoff p -Laplacian equations involving critical Hardy-Sobolev nonlinearities and nonnegative potentials.In [9] a multiplicity result for a fractional Kirchhoff equation involving a Beresticky-Lions type nonlinearity isproved. The author and Isernia [10] used penalization method and Lusternik-Schnirelmann category theoryto study the existence and multiplicity of solutions for a fractional Schrödinger-Kirchhoff equation withsubcritical nonlinearities; see also [29] in which the authors used the approach in [10] to consider a subcriticalversion of (1.1). Liu et al. [34], via the monotonicity trick and the profile decomposition, proved the existenceof ground states to a fractional Kirchhoff equation with critical nonlinearity in low dimension.Motivated by the above works, in this paper we aim to study the existence and concentration behavior ofsolutions to (1.1) under assumptions ( V ) - ( V ) and ( f ) - ( f ) . More precisely, our main result can be statedas follows: Theorem 1.1.
Assume that ( V ) - ( V ) and ( f ) - ( f ) hold. Then, there exists ε > such that, for each ε ∈ (0 , ε ) , problem (1.1) has a positive solution u ε . Moreover, if η ε denotes a global maximum point of u ε ,then we have lim ε → V ( η ε ) = V , and there exists a constant C > such that < u ε ( x ) ≤ Cε s ε s + | x − η ε | s for all x ∈ R . The proof of Theorem 1.1 will be done via appropriate variational arguments. After considering the ε -rescaled problem associated with (1.1), we use a variant of the penalization technique introduced in [17] (seealso [3,22]) which consists in modifying in a suitable way the nonlinearity outside Λ , solving a modified problemand then check that, for ε > small enough, the solutions of the modified problem are indeed solutions of theoriginal one. These solutions will be obtained as critical points of the modified energy functional J ε which, inview of the growth assumptions on f and the auxiliary nonlinearity, possesses a mountain pass geometry [5].In order to recover some compactness properties for J ε , we have to circumvent several difficulties which makeour study rather delicate. The first one is related to the presence of the Kirchhoff term in (1.1) which doesnot permit to verify in a standard way that if u is the weak limit of a Palais-Smale sequence ( ( P S ) in short) { u n } n ∈ N for J ε , then u is a weak solution for the modified problem. The second one is due to the lack ofcompactness caused by the unboundedness of the domain R and the critical Sobolev exponent. Anyway, wewill be able to overcome these problems looking for critical points of a suitable functional whose quadraticpart involves the limit term of ( a + b [ u n ] s ) , and showing that the mountain pass level c ε of J ε is strictly lessthan a threshold value related to the best constant of the embedding H s ( R ) in L ∗ s ( R ) . Then, applyingmountain pass lemma, we will deduce the existence of a positive solution for the modified problem. Finally,combining a compactness argument with a Moser iteration procedure [37], we prove that the solution of themodified problem is also a solution to the original one for ε > small enough, and that it decays at zero atinfinity with polynomial rate. To our knowledge, this is the first time that concentration phenomenon forproblem (1.1) is investigated in the literature.The paper is organized as follows: in Section we introduce the modified problem and we provide sometechnical results. In Section we give the proof of Theorem 1.1.2. The modified problem
Preliminaries.
Here we fix the notations and we recall some useful preliminary results on fractional Sobolev spaces (seealso [18, 36] for more details).If A ⊂ R , we denote by | u | L q ( A ) the L q ( A ) -norm of a function u : R → R , and by | u | q its L q ( R ) -norm. Wedenote by B r ( x ) the ball centered at x ∈ R with radius r > . When x = 0 , we put B r = B r (0) . Let usdefine D s, ( R ) as the completion of C ∞ c ( R ) with respect to the norm [ u ] s := Z Z R | u ( x ) − u ( y ) | | x − y | s dxdy = Z R | ( − ∆) s u | dx, where the second identity holds up to a constant; see [18]. Then we consider the fractional Sobolev space H s ( R ) := n u ∈ L ( R ) : [ u ] s < ∞ o endowed with the norm k u k := [ u ] s + | u | . V. AMBROSIO
We recall the following main embeddings for the fractional Sobolev spaces:
Theorem 2.1. [18] Let s ∈ (0 , . Then there exists a sharp constant S ∗ = S ∗ ( s ) > such that for any u ∈ D s, ( R ) | u | ∗ s ≤ S − ∗ [ u ] s . Moreover, H s ( R ) is continuously embedded in L p ( R ) for any p ∈ [2 , ∗ s ] and compactly in L ploc ( R ) for any p ∈ [1 , ∗ s ) . The following lemma is a version of the well-known Lions type result:
Lemma 2.1. [21] If { u n } n ∈ N is a bounded sequence in H s ( R ) and if lim n →∞ sup y ∈ R Z B R ( y ) | u n | dx = 0 for some R > , then u n → in L r ( R ) for all r ∈ (2 , ∗ s ) . We also recall the following useful technical result.
Lemma 2.2. [39] Let u ∈ D s, ( R ) . Let ϕ ∈ C ∞ c ( R ) and for each r > we define ϕ r ( x ) = ϕ ( x/r ) . Then, [ uϕ r ] s → as r → . If in addition ϕ = 1 in a neighborhood of the origin, then [ uϕ r ] s → [ u ] s as r → ∞ . Functional Setting.
In order to study (1.1), we use the change of variable x ε x and we will look for solutions to (cid:26) ( a + b [ u ] s )( − ∆) s u + V ( ε x ) u = f ( u ) + | u | ∗ s − u in R ,u ∈ H s ( R ) , u > in R . (2.1)Now, we introduce a penalization method in the spirit of [17] which will be fundamental to obtain ourmain result. First of all, without loss of generality, we will assume that ∈ Λ and V (0) = V = inf Λ V. Let
K > ϑϑ − and a > be such that f ( a ) + a ∗ s − = V K a (2.2)and we define ˜ f ( t ) := (cid:26) f ( t ) + ( t + ) ∗ s − if t ≤ a , V K t if t > a , and g ( x, t ) := (cid:26) χ Λ ( x )( f ( t ) + ( t + ) ∗ s − ) + (1 − χ Λ ( x )) ˜ f ( t ) if t > , if t ≤ . It is easy to check that g satisfies the following properties: ( g ) lim t → + g ( x,t ) t = 0 uniformly with respect to x ∈ R , ( g ) g ( x, t ) ≤ f ( t ) + t ∗ s − for all x ∈ R , t > , ( g ) ( i ) 0 ≤ ϑG ( x, t ) < g ( x, t ) t for all x ∈ Λ and t > , ( ii ) 0 ≤ G ( x, t ) < g ( x, t ) t ≤ V K t for all x ∈ R \ Λ and t > , ( g ) for each x ∈ Λ the function g ( x,t ) t is increasing in (0 , ∞ ) , and for each x ∈ R \ Λ the function g ( x,t ) t isincreasing in (0 , a ) .Then, we consider the following modified problem (cid:26) ( a + b [ u ] s )( − ∆) s u + V ( ε x ) u = g ( ε x, u ) in R ,u ∈ H s ( R ) , u > in R . (2.3)The corresponding energy functional is given by J ε ( u ) = 12 k u k ε + b u ] s − Z R G ( ε x, u ) dx, which is well-defined on the space H ε := (cid:26) u ∈ H s ( R ) : Z R V ( ε x ) u dx < ∞ (cid:27) RACTIONAL CRITICAL KIRCHHOFF EQUATIONS 5 endowed with the norm k u k ε := a [ u ] s + Z R V ( ε x ) u dx. Clearly H ε is a Hilbert space with the following inner product ( u, v ) ε := a Z Z R ( u ( x ) − u ( y ))( v ( x ) − v ( y )) | x − y | s dxdy + Z R V ( ε x ) uv dx. It is standard to show that J ε ∈ C ( H ε , R ) and its differential is given by hJ ′ ε ( u ) , v i = ( u, v ) ε + b [ u ] s Z Z R ( u ( x ) − u ( y ))( v ( x ) − v ( y )) | x − y | s dxdy − Z R g ( ε x, u ) v dx for any u, v ∈ H ε . Let us introduce the Nehari manifold associated with (2.3), that is, N ε := n u ∈ H ε \ { } : hJ ′ ε ( u ) , u i = 0 o . We begin by proving that J ε possesses a nice geometric structure: Lemma 2.3.
The functional J ε has a mountain-pass geometry: ( a ) there exist α, ρ > such that J ε ( u ) ≥ α with k u k ε = ρ ; ( b ) there exists e ∈ H ε with k e k ε > ρ such that J ε ( e ) < .Proof. ( a ) By assumptions ( g ) and ( g ) we deduce that for any ξ > there exists C ξ > such that J ε ( u ) ≥ k u k ε − Z R G ( ε x, u ) dx ≥ k u k ε − ξC k u k ε − C ξ C k u k ∗ s ε . Then, there exist α, ρ > such that J ε ( u ) ≥ α with k u k ε = ρ . ( b ) Using ( g ) - ( i ) , we deduce that for any u ∈ C ∞ c ( R ) \ { } such that u ≥ and supp ( u ) ⊂ Λ ε , and for all τ > it holds J ε ( τ u ) = τ k u k ε + b τ u ] s − Z Λ ε G ( ε x, τ u ) dx ≤ τ k u k ε + b τ u ] s − C τ ϑ Z Λ ε u ϑ dx + C , (2.4)for some constants C , C > . Recalling that ϑ ∈ (4 , ∗ s ) we can conclude that J ε ( τ u ) → −∞ as τ → ∞ . (cid:3) In view of Lemma 2.3, we can use a variant of the mountain-pass theorem without ( P S ) -condition (see [49])to deduce the existence of a Palais-Smale sequence { u n } n ∈ N ⊂ H ε such that J ε ( u n ) = c ε + o n (1) and J ′ ε ( u n ) = o n (1) (2.5)where c ε := inf γ ∈ Γ ε max t ∈ [0 , J ε ( γ ( t )) and Γ ε := n γ ∈ C ([0 , , H ε ) : γ (0) = 0 , J ε ( γ (1)) ≤ o . (2.6)As in [49], we can use the following equivalent characterization of c ε more appropriate for our aim: c ε = inf u ∈H ε \{ } max t ≥ J ε ( tu ) . Moreover, from the monotonicity of g , it is easy to see that for all u ∈ H ε \ { } there exists a unique t = t ( u ) > such that J ε ( t u ) = max t ≥ J ε ( tu ) . In the next lemma, we will see that c ε is less then a threshold value involving the best constant S ∗ of Sobolevembedding D s, ( R ) in L ∗ s ( R ) . More precisely: Lemma 2.4.
There exists
T > such that c ε < a S ∗ T − s + b S ∗ T − s − ∗ s T =: c ∗ for all ε > . V. AMBROSIO
Proof.
We argue as in [34]. Let η ∈ C ∞ c ( R ) be a cut-off function such that η = 1 in B ρ , supp( η ) ⊂ B ρ and ≤ η ≤ , where B ρ ⊂ Λ ε . For simplicity, we assume that ρ = 1 . We know (see [15]) that S ∗ is achieved by U ( x ) = κ ( µ + | x − x | ) − − s , with κ ∈ R , µ > and x ∈ R . Taking x = 0 , as in [45], we can define v h ( x ) := η ( x ) u h ( x ) ∀ h > , where u h ( x ) := h − − s u ∗ ( x/h ) and u ∗ ( x ) := U ( x/S s ∗ ) | U | ∗ s . Then ( − ∆) s u h = | u h | ∗ s − u h in R and [ u h ] s = | u h | ∗ s ∗ s = S s ∗ . We also recall the following useful estimates: A h := [ v h ] s = S s ∗ + O ( h − s ) (2.7) B h := | v h | = O ( h − s ) (2.8) C h := | v h | qq ≥ O ( h − (3 − s ) q ) if q > − s O (log( h ) h − (3 − s ) q ) if q = − s O ( h (3 − s ) q ) if q < − s (2.9) D h := | v h | ∗ s ∗ s = S s ∗ + O ( h ) . (2.10)Let us note that for all h > there exists t > such that J ε ( γ h ( t )) < , where γ h ( t ) = v h ( · /t ) . Indeed,setting V := max x ∈ Λ V ( x ) , by ( f ) we have J ε ( γ h ( t )) ≤ a t − s [ v h ] s + V t | v h | + b t − s [ v h ] s − t ∗ s | v h | ∗ s ∗ s − t q | v h | qq C = a t − s A h + b A h t − s + (cid:18) V B h − D h ∗ s − C C h q (cid:19) t . (2.11)Since < − s < , we can use (2.8) to deduce that V B h − D h ∗ s → − ∗ s S s ∗ as h → . Hence, using (2.7), we can see that for all h > sufficiently small J ε ( γ h ( t )) → −∞ as t → ∞ ,that is there exists t > such that J ε ( γ h ( t )) < .Now, as t → + , we have [ γ h ( t )] s + | γ h ( t ) | = t − s A h + t B h → uniformly for h > small.We set γ h (0) = 0 . Then γ h ( t · ) ∈ Γ ε , where Γ ε is defined as in (2.6) and we infer that c ε ≤ sup t ≥ J ε ( γ h ( t )) . Taking into account that c ε > , by (2.11) there exists t h > such that sup t ≥ J ε ( γ h ( t )) = J ε ( γ h ( t h )) . In the light of (2.7), (2.9) and (2.11) we deduce that J ε ( γ h ( t )) → + as t → + and J ε ( γ h ( t )) → −∞ as t → ∞ uniformly for h > small. Then there exist t , t > (independent of h > ) satisfying t ≤ t h ≤ t .Set H h ( t ) := aA h t − s + bA h t − s − D h ∗ s t . Therefore, c ε ≤ sup t ≥ H h ( t ) + (cid:18) V B h − C C h q (cid:19) t h . From (2.9), for any q ∈ (2 , ∗ s ) , we have C h ≥ O ( h − (3 − s ) q ) . Then, by (2.8), we can infer c ε ≤ sup t ≥ H h ( t ) + O ( h − s ) − O ( C h − (3 − s ) q ) . RACTIONAL CRITICAL KIRCHHOFF EQUATIONS 7
Since − s > and − (3 − s ) q > , we obtain sup t ≥ H h ( t ) ≥ c ε uniformly for h > small.Arguing as above, there exist t , t > (independent of h > ) such that sup t ≥ H h ( t ) = sup t ∈ [ t ,t ] H h ( t ) . By (2.7) we deduce c ε ≤ sup t ≥ K ( S s ∗ t ) + O ( h − s ) − O ( C h − (3 − s ) q ) , (2.12)where K ( t ) := aS s t − s + bS s t − s − ∗ s t . Let us note that for t > , K ′ ( t ) = 3 − s aS ∗ t − s + 3 − s bS ∗ t − s − − s t = (3 − s ) t − s (cid:0) aS ∗ + bS ∗ t − s − t s (cid:1) =: (3 − s ) t − s K ( t ) . Moreover, ˜ K ′ ( t ) = bS ∗ (3 − s ) t − s − st s − = t − s [ bS ∗ (3 − s ) − st s − ] . Since s > , there exists a unique T > such that ˜ K ( t ) > for t ∈ (0 , T ) and ˜ K ( t ) < for t > T . Thus, T is the unique maximum point of K ( t ) . In virtue of (2.12) we have c ε ≤ K ( T ) + O ( h − s ) − O ( C h − (3 − s ) q ) . (2.13)If q > s − s , then < − (3 − s ) q < − s , and by (2.13), for any fixed C > , it holds c ε < K ( T ) for h > small. If < q < s − s , then, for h > small and C > h (3 − s ) q − s − , we also have c ε < K ( T ) . (cid:3) Lemma 2.5.
Every sequence { u n } n ∈ N satisfying (2.5) is bounded in H ε .Proof. In view of ( g ) we can deduce that c ε + o n (1) k u n k ε ≥ J ε ( u n ) − ϑ hJ ′ ε ( u n ) , u n i (2.14) = (cid:18) ϑ − ϑ (cid:19) k u n k ε + b (cid:18) ϑ − ϑ (cid:19) [ u n ] + 1 ϑ Z R \ Λ ε [ g ( ε x, u n ) u n − ϑG ( ε x, u n )] dx + 1 ϑ Z Λ ε [ g ( ε x, u n ) u n − ϑG ( ε x, u n )] dx ≥ (cid:18) ϑ − ϑ (cid:19) k u n k ε + 1 ϑ Z R \ Λ ε [ g ( ε x, u n ) u n − ϑG ( ε x, u n )] dx ≥ (cid:18) ϑ − ϑ (cid:19) k u n k ε − (cid:18) ϑ − ϑ (cid:19) K Z R \ Λ ε V ( ε x ) u n dx ≥ (cid:18) ϑ − ϑ (cid:19) (cid:18) − K (cid:19) k u n k ε . (2.15)Since ϑ > and K > , we can conclude that { u n } n ∈ N is bounded in H ε . (cid:3) Lemma 2.6.
There exist a sequence { z n } n ∈ N ⊂ R and R, β > such that Z B R ( z n ) u n dx ≥ β. Moreover, { z n } n ∈ N is bounded in R . V. AMBROSIO
Proof.
Assume by contradiction that the first conclusion of lemma is not true. From Lemma 2.1 we have u n → in L q ( R ) ∀ q ∈ (2 , ∗ s ) , which together with ( f ) and ( f ) yields Z R F ( u n ) dx = Z R f ( u n ) u n dx = o n (1) as n → ∞ . Since { u n } n ∈ N is bounded in H ε , we may assume that u n ⇀ u in H ε .Now, we can observe that Z R G ( ε x, u n ) dx ≤ ∗ s Z Λ ε ∪{ u n ≤ a } ( u + n ) ∗ s dx + V K Z ( R \ Λ ε ) ∩{ u n >a } u n dx + o n (1) (2.16)and Z R g ( ε x, u n ) u n dx = Z Λ ε ∪{ u n ≤ a } ( u + n ) ∗ s dx + V K Z ( R \ Λ ε ) ∩{ u n >a } u n dx + o n (1) . (2.17)Using hJ ′ ε ( u n ) , u n i = o n (1) and (2.17) we have k u n k ε − V K Z ( R \ Λ ε ) ∩{ u n >a } u n dx + b [ u n ] s = Z Λ ε ∪{ u n ≤ a } ( u + n ) ∗ s dx + o n (1) . (2.18)Assume that Z Λ ε ∪{ u n ≤ a } ( u + n ) ∗ s dx → ℓ ≥ and [ u n ] s → B . Note that ℓ > , otherwise (2.18) yields k u n k ε → as n → ∞ which implies that J ε ( u n ) → , and this isimpossible because c ε > . Then, by (2.18) and the Sobolev inequality we obtain aS ∗ Z Λ ε ∪{ u n ≤ a } ( u + n ) ∗ s dx ! ∗ s + bS ∗ Z Λ ε ∪{ u n ≤ a } ( u + n ) ∗ s dx ! ∗ s ≤ Z Λ ε ∪{ u n ≤ a } ( u + n ) ∗ s dx + o n (1) . (2.19)Since ℓ > , it follows from (2.19) that K ′ ( ℓ ) = 3 − s ℓ − ( aS ∗ ℓ − s + bS ∗ ℓ − s − ℓ ) ≤ so we can deduce that ℓ ≥ T , where T is the unique maximum of K defined in Lemma 2.4.Let us consider the following functional: I ε ( u ) := ( a + bB )2 [ u ] s + 12 Z R V ( ε x ) u dx − Z R G ( ε x, u ) dx = J ε ( u ) − b u ] s + b B [ u ] s , (2.20)and we note that { u n } n ∈ N is a ( P S ) c ε + b B sequence for I ε , that is I ε ( u n ) = c ε + b B + o n (1) , I ′ ε ( u n ) = o n (1) . (2.21) RACTIONAL CRITICAL KIRCHHOFF EQUATIONS 9
Then, using (2.16), (2.21), ℓ ≥ T and the Sobolev inequality we can infer c ε = I ε ( u n ) − b B + o n (1) ≥ a u n ] s + bB u n ] s − b B + 12 Z R V ( ε x ) u n dx − V K Z ( R \ Λ ε ) ∩{ u n >a } u n dx − ∗ s Z Λ ε ∪{ u n ≤ a } ( u + n ) ∗ s dx + o n (1) ≥ a u n ] s + b u n ] s − ∗ s Z Λ ε ∪{ u n ≤ a } ( u + n ) ∗ s dx + o n (1) ≥ a S ∗ Z Λ ε ∪{ u n ≤ a } ( u + n ) ∗ s dx ! ∗ s + b S ∗ Z Λ ε ∪{ u n ≤ a } ( u + n ) ∗ s dx ! ∗ s − ∗ s Z Λ ε ∪{ u n ≤ a } ( u + n ) ∗ s dx + o n (1)= a S ∗ ℓ − s + b S ∗ ℓ − s − ∗ s ℓ ≥ a S ∗ T − s + b S ∗ T − s − ∗ s T = c ∗ , and this gives a contradiction by Lemma 2.4.Now, we show that { z n } n ∈ N is bounded in R . For any ρ > , let ψ ρ ∈ C ∞ ( R ) be such that ψ ρ = 0 in B ρ and ψ ρ = 1 in R \ B ρ , with ≤ ψ ρ ≤ and |∇ ψ ρ | ≤ Cρ , where C is a constant independent of ρ . Since { ψ ρ u n } n ∈ N is bounded in H ε , it follows that hJ ′ ε ( u n ) , ψ ρ u n i = o n (1) , that is ( a + b [ u n ] s ) Z Z R | u n ( x ) − u n ( y ) | | x − y | s ψ ρ ( x ) dxdy + Z R V ( ε x ) u n ψ ρ dx = o n (1) + Z R g ( ε x, u n ) u n ψ ρ dx − ( a + b [ u n ] s ) Z Z R ( ψ ρ ( x ) − ψ ρ ( y ))( u n ( x ) − u n ( y )) | x − y | s u n ( y ) dxdy. Take ρ > such that Λ ε ⊂ B ρ . Then, using ( g ) - ( ii ) , we get Z Z R a | u n ( x ) − u n ( y ) | | x − y | s ψ ρ ( x ) dxdy + Z R V ( ε x ) u n ψ ρ dx ≤ Z R K V ( ε x ) u n ψ ρ dx − ( a + b [ u n ] s ) Z Z R ( ψ ρ ( x ) − ψ ρ ( y ))( u n ( x ) − u n ( y )) | x − y | s u n ( y ) dxdy + o n (1) which implies that (cid:18) − K (cid:19) V Z R u n ψ ρ dx ≤ − ( a + b [ u n ] s ) Z Z R ( ψ ρ ( x ) − ψ ρ ( y ))( u n ( x ) − u n ( y )) | x − y | s u n ( y ) dxdy + o n (1) . (2.22)Now, from the Hölder inequality and the boundedness on { u n } n ∈ N in H ε we can see that (cid:12)(cid:12)(cid:12)(cid:12)Z Z R ( u n ( x ) − u n ( y ))( ψ ρ ( x ) − ψ ρ ( y )) | x − y | s u n ( y ) dxdy (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18)Z Z R | ψ ρ ( x ) − ψ ρ ( y ) | | x − y | s | u n ( y ) | dxdy (cid:19) . (2.23) On the other hand, recalling that ≤ ψ ρ ≤ and |∇ ψ ρ | ∞ ≤ C/ρ and using polar coordinates, we obtain
Z Z R | ψ ρ ( x ) − ψ ρ ( y ) | | x − y | s | u n ( x ) | dxdy = Z R Z | y − x | >ρ | ψ ρ ( x ) − ψ ρ ( y ) | | x − y | s | u n ( x ) | dxdy + Z R Z | y − x |≤ ρ | ψ ρ ( x ) − ψ ρ ( y ) | | x − y | s | u n ( x ) | dxdy ≤ C Z R | u n ( x ) | Z | y − x | >ρ dy | x − y | s ! dx + Cρ Z R | u n ( x ) | Z | y − x |≤ ρ dy | x − y | s − ! dx ≤ C Z R | u n ( x ) | Z | z | >ρ dz | z | s ! dx + Cρ Z R | u n ( x ) | Z | z |≤ ρ dz | z | s ! dx ≤ C Z R | u n ( x ) | dx (cid:18)Z ∞ ρ dρρ s +1 (cid:19) + Cρ Z R | u n ( x ) | dx (cid:18)Z ρ dρρ s − (cid:19) ≤ Cρ s Z R | u n ( x ) | dx + Cρ ρ − s +2 Z R | u n ( x ) | dx ≤ Cρ s Z R | u n ( x ) | dx ≤ Cρ s where in the last passage we used the boundedness of { u n } n ∈ N in H ε . Taking into account (2.22), (2.23) andthe above estimate we can infer that (cid:18) − K (cid:19) V Z R u n ψ ρ dx ≤ Cρ s + o n (1) which implies that { z n } n ∈ N is bounded in R . (cid:3) We conclude this section giving the proof of the main result of this section:
Theorem 2.2.
Assume that ( V ) - ( V ) and ( f - ( f hold. Then, problem (2.3) admits a positive groundstate for all ε > .Proof. Using Lemma 2.3 and a variant of the mountain pass theorem without ( P S ) condition (see [49]), weknow that there exists a Palais-Smale sequence { u n } n ∈ N for J ε at the level c ε , where c ε < c ∗ by Lemma2.4. Taking into account Lemma 2.5, we can see that { u n } n ∈ N is bounded in H ε , so we may assume that u n ⇀ u in H ε and u n → u in L qloc ( R ) for all q ∈ [1 , ∗ s ) . It follows from Lemma 2.6 that u nontrivial. Since hJ ′ ε ( u n ) , ϕ i = o n (1) for all ϕ ∈ H ε , we can see that Z R a ( − ∆) s u ( − ∆) s ϕ + V ( ε x ) uϕ dx + bB (cid:18)Z R ( − ∆) s u ( − ∆) s ϕ dx (cid:19) = Z R g ( ε x, u ) ϕ dx, (2.24)where B := lim n →∞ [ u n ] s . Let us note that B ≥ [ u ] s by Fatou’s Lemma. If by contradiction B > [ u ] s , wemay use (2.24) to deduce that hJ ′ ε ( u ) , u i < . Moreover, conditions ( g ) - ( g ) imply that hJ ′ ε ( τ u ) , τ u i > forsome < τ << . Then there exists t ∈ ( τ, such that t u ∈ N ε and hJ ′ ε ( t u ) , t u i = 0 . Using Fatou’sLemma, t ∈ ( τ, and ( g ) we get c ε ≤ J ε ( t u ) − hJ ′ ε ( t u ) , t u i < J ε ( u ) − hJ ′ ε ( u ) , u i ≤ lim inf n →∞ (cid:20) J ε ( u n ) − hJ ′ ε ( u n ) , u n i (cid:21) = c ε (2.25)which gives a contradiction. Therefore B = [ u ] s and we deduce that J ′ ε ( u ) = 0 . Hence, J ε admits anontrivial critical point u ∈ H ε . Since hJ ′ ε ( u ) , u − i = 0 , where u − = min { u, } , and g ( x, t ) = 0 for t ≤ , it iseasy to check that u ≥ in R . Moreover, proceeding as in the proof of Lemma 3.2 below, we can see that u ∈ L ∞ ( R ) . By Proposition . in [47] and s > we deduce that u ∈ C ,α ( R ) , and applying the maximumprinciple [47] we can conclude that u > in R . Finally, arguing as in (2.25) with t = 1 , we can show that u is a ground state solution to (2.3). (cid:3) The limiting problem.
Let us consider the following limiting problem related to (2.3), that is, for µ > (cid:26) ( a + b [ u ] s )( − ∆) s u + µu = f ( u ) + | u | ∗ s − u in R ,u ∈ H s ( R ) , u > in R , (2.26) RACTIONAL CRITICAL KIRCHHOFF EQUATIONS 11 whose corresponding Euler-Lagrange functional is given by I µ ( u ) = 12 (cid:0) a [ u ] s + µ | u | (cid:1) + b u ] s − Z R F ( u ) + 12 ∗ s ( u + ) ∗ s dx which is well defined on the Hilbert space H µ := H s ( R ) endowed with the inner product ( u, ϕ ) µ := a Z Z R ( u ( x ) − u ( y ))( ϕ ( x ) − ϕ ( y )) | x − y | s dxdy + µ Z R u ( x ) ϕ ( x ) dx. The norm induced by the above inner product is given by k u k µ := a [ u ] s + µ | u | . We denote by M µ the Nehari manifold associated with I µ , that is M µ := n u ∈ H µ \ { } : hI ′ µ ( u ) , u i = 0 o , and d µ := inf u ∈M µ I µ ( u ) , or equivalently d µ = inf u ∈H µ \{ } max t ≥ I µ ( tu ) . Arguing as in the proof of Theorem 2.2, it is easy to deduce that:
Theorem 2.3.
For all µ > , problem (2.26) admits a positive ground state solution. Let us prove the following useful relation between c ε and d V : Lemma 2.7.
It holds lim sup ε → c ε ≤ d V .Proof. For any ε > we set ω ε ( x ) := ψ ε ( x ) ω ( x ) , where ω is a positive ground state given by Theorem 2.3with µ = V , and ψ ε ( x ) := ψ ( ε x ) with ψ ∈ C ∞ c ( R ) , ψ ∈ [0 , , ψ ( x ) = 1 if | x | ≤ and ψ ( x ) = 0 if | x | ≥ .Here we assume that supp ( ψ ) ⊂ B ⊂ Λ . Using Lemma 2.2 and the dominated convergence theorem we cansee that ω ε → ω in H s ( R ) and I V ( ω ε ) → I V ( ω ) = d V as ε → . For each ε > there exists t ε > suchthat J ε ( t ε ω ε ) = max t ≥ J ε ( tω ε ) . Then, J ′ ε ( t ε ω ε ) = 0 and this implies that t ε Z R a | ( − ∆) s ω ε | + V ( ε x ) ω ε dx + b (cid:18)Z R | ( − ∆) s ω ε | dx (cid:19) = Z R f ( t ε ω ε )( t ε ω ε ) ω ε dx + t ∗ s − ε Z R | ω ε | ∗ s dx. (2.27)By ( f ) - ( f ) , ω ∈ M V and (2.27) it follows that t ε → as ε → . On the other hand, c ε ≤ max t ≥ J ε ( tω ε ) = J ε ( t ε ω ε ) = I V ( t ε ω ε ) + t ε Z R ( V ( ε x ) − V ) ω ε dx. Since V ( ε x ) is bounded on the support of ω ε , by the dominated convergence theorem and the above inequality,we obtain the thesis. (cid:3) Proof of Theorem 1.1
This last section is devoted to the proof of the main result of this work. Firstly, we prove the followingcompactness result which will be fundamental to show that the solutions of (2.3) are also solutions to (2.1)for ε > small enough. Lemma 3.1.
Let ε n → + and { u n } n ∈ N := { u ε n } n ∈ N ⊂ H ε n be such that J ε n ( u n ) = c ε n and J ′ ε n ( u n ) = 0 .Then there exists { ˜ y n } n ∈ N ⊂ R such that the translated sequence ˜ u n ( x ) := u n ( x + ˜ y n ) has a subsequence which converges in H s ( R ) . Moreover, up to a subsequence, { y n } n ∈ N := { ε n ˜ y n } n ∈ N is suchthat y n → y for some y ∈ Λ such that V ( y ) = V . Proof.
Using hJ ′ ε n ( u n ) , u n i = 0 and ( g ) , ( g ) , it is easy to see that there is γ > (independent of ε n ) suchthat k u n k ε n ≥ γ > ∀ n ∈ N . Taking into account J ε n ( u n ) = c ε n , hJ ′ ε n ( u n ) , u n i = 0 and Lemma 2.7, we can argue as in the proof ofLemma 2.5 to deduce that { u n } n ∈ N is bounded in H ε n . Therefore, proceeding as in Lemma 2.6, we can finda sequence { ˜ y n } n ∈ N ⊂ R and constants R, α > such that lim inf n →∞ Z B R (˜ y n ) | u n | dx ≥ α. Set ˜ u n ( x ) := u n ( x + ˜ y n ) . Then, { ˜ u n } n ∈ N is bounded in H s ( R ) , and we may assume that ˜ u n ⇀ ˜ u weakly in H s ( R ) , (3.1)and [˜ u n ] s → B as n → ∞ . Moreover, ˜ u = 0 in view of Z B R | ˜ u | dx ≥ α. (3.2)Now, we set y n := ε n ˜ y n . Firstly, we show that { y n } n ∈ N is bounded. To achieve our purpose, we prove thefollowing claim: Claim 1 lim n →∞ dist ( y n , Λ) = 0 .If by contradiction the claim is not true, then we can find δ > and a subsequence of { y n } n ∈ N , still denotedby itself, such that dist ( y n , Λ) ≥ δ ∀ n ∈ N . Thus, there is r > such that B r ( y n ) ⊂ R \ Λ for all n ∈ N . Since ˜ u ≥ and C ∞ c ( R ) is dense in H s ( R ) ,we can approximate ˜ u by a sequence { ψ j } j ∈ N ⊂ C ∞ c ( R ) such that ψ j ≥ in R , so that ψ j → ˜ u in H s ( R ) .Fix j ∈ N and use ψ = ψ j as test function in hJ ′ ε n ( u n ) , ψ i = 0 . Then we have ( a + b [˜ u n ] s ) Z Z R (˜ u n ( x ) − ˜ u n ( y ))( ψ j ( x ) − ψ j ( y )) | x − y | s dxdy + Z R V ( ε n x + ε n ˜ y n )˜ u n ψ j dx = Z R g ( ε n x + ε n ˜ y n , ˜ u n ) ψ j dx. (3.3)Since u ε n , ψ j ≥ and using the definition of g , we can note that Z R g ( ε n x + ε n ˜ y n , ˜ u n ) ψ j dx = Z B r/ εn g ( ε n x + ε n ˜ y n , ˜ u n ) ψ j dx + Z R \B r/ εn g ( ε n x + ε n ˜ y n , ˜ u n ) ψ j dx ≤ V K Z B r/ εn ˜ u n ψ j dx + Z R \B r/ εn (cid:16) f (˜ u n ) ψ j + ˜ u ∗ s − n ψ j (cid:17) dx. This fact together with (3.3) gives ( a + b [˜ u n ] s ) Z Z R (˜ u n ( x ) − ˜ u n ( y ))( ψ j ( x ) − ψ j ( y )) | x − y | s dxdy + A Z R ˜ u n ψ j dx ≤ Z R \B r/ εn (cid:16) f (˜ u n ) ψ j + ˜ u ∗ s − n ψ j (cid:17) dx (3.4)where A = V (1 − K ) . Taking into account (3.1), ψ j has compact support in R and ε n → + , we can inferthat as n → ∞ Z Z R (˜ u n ( x ) − ˜ u n ( y ))( ψ j ( x ) − ψ j ( y )) | x − y | s dxdy → Z Z R (˜ u ( x ) − ˜ u ( y ))( ψ j ( x ) − ψ j ( y )) | x − y | s dxdy and Z R \B r/ εn (cid:16) f (˜ u n ) ψ j + ˜ u ∗ s − n ψ j (cid:17) dx → . The above limits, (3.4) and [˜ u n ] s → B imply that ( a + bB ) Z Z R (˜ u ( x ) − ˜ u ( y ))( ψ j ( x ) − ψ j ( y )) | x − y | s dxdy + A Z R ˜ uψ j dx ≤ RACTIONAL CRITICAL KIRCHHOFF EQUATIONS 13 and passing to the limit as j → ∞ we can infer that ( a + bB )[˜ u ] s + A | ˜ u | ≤ . This gives a contradiction by (3.2). Hence, there exists a subsequence of { y n } n ∈ N such that y n → y ∈ Λ . Secondly, we prove the following claim:
Claim 2 y ∈ Λ .In the light of ( g ) and (3.3) we can deduce that ( a + b [˜ u n ] s ) Z Z R (˜ u n ( x ) − ˜ u n ( y ))( ψ j ( x ) − ψ j ( y )) | x − y | s dxdy + Z R V ( ε n x + ε n ˜ y n )˜ u n ψ j dx ≤ Z R ( f (˜ u n ) + ˜ u ∗ s − n ) ψ j dx. Letting n → ∞ we find ( a + bB ) Z Z R (˜ u ( x ) − ˜ u ( y ))( ψ j ( x ) − ψ j ( y )) | x − y | s dxdy + Z R V ( y )˜ uψ j dx ≤ Z R ( f (˜ u ) + ˜ u ∗ s − ) ψ j dx, and passing to the limit as j → ∞ we obtain ( a + bB )[˜ u ] s + V ( y ) | ˜ u | ≤ Z R ( f (˜ u ) + ˜ u ∗ s − )˜ u dx. Since B ≥ [˜ u ] s (by Fatou’s Lemma), the above inequality yields ( a + b [˜ u ] s )[˜ u ] s + V ( y ) | ˜ u | ≤ Z R ( f (˜ u ) + ˜ u ∗ s − )˜ u dx. Therefore, we can find τ ∈ (0 , such that τ ˜ u ∈ M V ( y ) . Then, by Lemma 2.7, we can see that d V ( y ) ≤ I V ( y ) ( τ ˜ u ) ≤ lim inf n →∞ J ε n ( u n ) = lim inf n →∞ c ε n ≤ d V which implies that V ( y ) ≤ V (0) = V . Since V = min ¯Λ V , we can deduce that V ( y ) = V . This facttogether with ( V ) yields y / ∈ ∂ Λ . Consequently, y ∈ Λ . Claim 3 ˜ u n → ˜ u in H s ( R ) as n → ∞ .Let us define ˜Λ n := Λ − ε n ˜ y n ε n and ˜ χ n ( x ) := (cid:26) if x ∈ ˜Λ n , if x ∈ R \ ˜Λ n , ˜ χ n ( x ) := 1 − ˜ χ n ( x ) . Let us also consider the following functions for all x ∈ R h n ( x ) := (cid:18) − ϑ (cid:19) V ( ε n x + ε n ˜ y n ) | ˜ u n ( x ) | ˜ χ n ( x ) h ( x ) := (cid:18) − ϑ (cid:19) V ( y ) | ˜ u ( x ) | h n ( x ):= (cid:20)(cid:18) − ϑ (cid:19) V ( ε n x + ε n ˜ y n ) | ˜ u n ( x ) | + 1 ϑ g ( ε n x + ε n ˜ y n , ˜ u n ( x ))˜ u n ( x ) − G ( ε n x + ε n ˜ y n , ˜ u n ( x )) (cid:21) ˜ χ n ( x ) ≥ (cid:18)(cid:18) − ϑ (cid:19) − K (cid:19) V ( ε n x + ε n ˜ y n ) | ˜ u n ( x ) | ˜ χ n ( x ) h n ( x ) := (cid:18) ϑ g ( ε n x + ε n ˜ y n , ˜ u n ( x ))˜ u n ( x ) − G ( ε n x + ε n ˜ y n , ˜ u n ( x )) (cid:19) ˜ χ n ( x )= (cid:20) ϑ (cid:16) f (˜ u n ( x ))˜ u n ( x ) + | ˜ u n ( x ) | ∗ s (cid:17) − (cid:18) F (˜ u n ( x )) + 12 ∗ s | ˜ u n ( x ) | ∗ s (cid:19)(cid:21) ˜ χ n ( x ) h ( x ) := 1 ϑ (cid:16) f (˜ u ( x ))˜ u ( x ) + | ˜ u ( x ) | ∗ s (cid:17) − (cid:18) F (˜ u ( x )) + 12 ∗ s | ˜ u ( x ) | ∗ s (cid:19) . In view of ( f ) and ( g ) , we can observe that the above functions are nonnegative. Moreover, by (3.1) andClaim , we know that ˜ u n ( x ) → ˜ u ( x ) a.e. x ∈ R ,y n = ε n ˜ y n → y ∈ Λ , which imply that ˜ χ n ( x ) → , h n ( x ) → h ( x ) , h n ( x ) → and h n ( x ) → h ( x ) a.e. x ∈ R . Hence, applying Fatou’s Lemma and using the invariance of R by translation, we can see that d V ≥ lim sup n →∞ c ε n = lim sup n →∞ (cid:18) J ε n ( u n ) − ϑ hJ ′ ε n ( u n ) , u n i (cid:19) ≥ lim sup n →∞ (cid:20)(cid:18) − ϑ (cid:19) [˜ u n ] s + (cid:18) − ϑ (cid:19) b [˜ u n ] s + Z R ( h n + h n + h n ) dx (cid:21) ≥ lim inf n →∞ (cid:20)(cid:18) − ϑ (cid:19) [˜ u n ] s + (cid:18) − ϑ (cid:19) b [˜ u n ] s + Z R ( h n + h n + h n ) dx (cid:21) ≥ (cid:18) − ϑ (cid:19) [˜ u ] s + (cid:18) − ϑ (cid:19) b [˜ u ] s + Z R ( h + h ) dx ≥ d V . Accordingly lim n →∞ [˜ u n ] s = [˜ u ] s (3.5)and h n → h , h n → and h n → h in L ( R ) . Then lim n →∞ Z R V ( ε n x + ε n ˜ y n ) | ˜ u n | dx = Z R V ( y ) | ˜ u | dx, and we can deduce that lim n →∞ | ˜ u n | = | ˜ u | . (3.6)Putting together (3.1), (3.5) and (3.6) and using the fact that H s ( R ) is a Hilbert space we obtain k ˜ u n − ˜ u k V → as n → ∞ . This fact ends the proof of lemma. (cid:3)
In the next lemma, we use a Moser iteration argument [37] to prove the following useful L ∞ -estimate for thesolutions of the modified problem (2.3). RACTIONAL CRITICAL KIRCHHOFF EQUATIONS 15
Lemma 3.2.
Let ε n → and u n ∈ H ε n be a solution to (2.3) . Then, up to a subsequence, ˜ u n := u n ( · + ˜ y n ) ∈ L ∞ ( R ) , and there exists C > such that | ˜ u n | ∞ ≤ C for all n ∈ N . Proof.
For any
L > and β > , let us define the function γ (˜ u n ) := γ L,β (˜ u n ) = ˜ u n ˜ u β − L,n ∈ H ε where ˜ u L,n := min { ˜ u n , L } . Since γ is an increasing function, we have ( a − b )( γ ( a ) − γ ( b )) ≥ for any a, b ∈ R . Let us consider E ( t ) := | t | and Γ( t ) := Z t ( γ ′ ( τ )) dτ. Then, applying Jensen’s inequality we get for all a, b ∈ R such that a > b , E ′ ( a − b )( γ ( a ) − γ ( b )) = ( a − b )( γ ( a ) − γ ( b )) = ( a − b ) Z ab γ ′ ( t ) dt = ( a − b ) Z ab (Γ ′ ( t )) dt ≥ (cid:18)Z ab (Γ ′ ( t )) dt (cid:19) . The same argument works when a ≤ b . Therefore E ′ ( a − b )( γ ( a ) − γ ( b )) ≥ | Γ( a ) − Γ( b ) | for any a, b ∈ R . (3.7)From (3.7), we can see 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 )) . (3.8)Choosing γ (˜ u n ) = ˜ u n ˜ u β − L,n as test function in (2.3) and using (3.8) we obtain a [Γ(˜ u n )] s + Z R V n ( x ) | ˜ u n | ˜ u β − L,n dx ≤ ( a + b [˜ u n ] s ) Z Z R (˜ u n ( x ) − ˜ u n ( y )) | x − y | N +2 s ((˜ u n ˜ u β − L,n )( x ) − (˜ u n ˜ u β − L,n )( y )) dxdy + Z R V n ( x ) | ˜ u n | ˜ u β − L,n dx ≤ Z R g n (˜ u n )˜ u n ˜ u β − L,n dx, (3.9)where V n ( x ) := V ( ε n x + ε n ˜ y n ) and g n ( x ) := g ( ε n x + ε n ˜ y n , ˜ u n ) . Since Γ(˜ u n ) ≥ β ˜ u n ˜ u β − L,n , and by Theorem 2.1, we have [Γ(˜ u n )] s ≥ S ∗ | Γ(˜ u n ) | ∗ s ≥ (cid:18) β (cid:19) S ∗ | ˜ u n ˜ u β − L,n | ∗ s . (3.10)On the other hand, by assumptions ( g ) and ( g ) , for any ξ > there exists C ξ > such that | g n (˜ u n ) | ≤ ξ | ˜ u n | + C ξ | ˜ u n | ∗ s − . (3.11)Thus, taking ξ ∈ (0 , V ) , and from (3.10) and (3.11), we can see that (3.9) yields | w L,n | ∗ s ≤ Cβ Z R | ˜ u n | ∗ s ˜ u β − L,n dx. (3.12) where w L,n := ˜ u n ˜ u β − L,n . Now, we take β = ∗ s and fix R > . Recalling that ≤ ˜ u L,n ≤ ˜ u n , we have Z R ˜ u ∗ s n v β − L,n dx = Z R ˜ u ∗ s − n ˜ u n v ∗ s − L,n dx = Z R ˜ u ∗ s − n (˜ u n ˜ u ∗ s − L,n ) dx ≤ Z { ˜ u n
1) = 2 ∗ s β m and β = ∗ s . Then we have (cid:18)Z R ˜ u β m +1 ∗ s n dx (cid:19) ∗ s ( βm +1 − ≤ ( Cβ m +1 ) βm +1 − (cid:18)Z R ˜ u ∗ s β m n dx (cid:19) ∗ s ( βm − . Let us define D m := (cid:18)Z R ˜ u ∗ s β m n dx (cid:19) ∗ s ( βm − . A standard iteration argument shows that we can find C > independent of m such that D m +1 ≤ m Y k =1 ( Cβ k +1 ) βk +1 − D ≤ C D . Passing to the limit as m → ∞ we get | ˜ u n | ∞ ≤ K for all n ∈ N . (cid:3) Now, we give the proof of Theorem 1.1.
Proof of Theorem 1.1.
Firstly, we prove that there exists ˜ ε > such that for any ε ∈ (0 , ˜ ε ) and anymountain-pass solution u ε ∈ H ε of (2.3), it results | u ε | L ∞ ( R \ Λ ε ) < a . (3.15)Assume by contradiction that for some subsequence { ε n } n ∈ N such that ε n → + , we can find u n := u ε n ∈ H ε n such that J ε n ( u n ) = c ε n , J ′ ε n ( u n ) = 0 and | u n | L ∞ ( R \ Λ εn ) ≥ a . (3.16) RACTIONAL CRITICAL KIRCHHOFF EQUATIONS 17
From Lemma 3.1, there exists { ˜ y n } n ∈ N ⊂ R such that ˜ u n = u n ( · + ˜ y n ) → ˜ u in H s ( R ) and ε n ˜ y n → y forsome y ∈ Λ such that V ( y ) = V . Now, if we choose r > such that B r ( y ) ⊂ B r ( y ) ⊂ Λ , we can see that B rεn ( y ε n ) ⊂ Λ ε n . Then, for any y ∈ B rεn (˜ y n ) it holds (cid:12)(cid:12)(cid:12)(cid:12) y − y ε n (cid:12)(cid:12)(cid:12)(cid:12) ≤ | y − ˜ y n | + (cid:12)(cid:12)(cid:12)(cid:12) ˜ y n − y ε n (cid:12)(cid:12)(cid:12)(cid:12) < ε n ( r + o n (1)) < rε n for n sufficiently large.Hence, for these values of n we have R \ Λ ε n ⊂ R \ B rεn (˜ y n ) . (3.17)Now, we observe that ˜ u n is a solution to ( − ∆) s ˜ u n + ˜ u n = ξ n in R , where ξ n ( x ) := ( a + b [˜ u n ] s ) − ( g n − V n ˜ u n ) + ˜ u n and V n ( x ) := V ( ε n x + ε n ˜ y n ) and g n ( x ) := g ( ε n x + ε n ˜ y n , ˜ u n ) . Put ξ ( x ) := ( a + b [˜ u ] s ) − [ f (˜ u ) + | ˜ u | ∗ s − ˜ u − V ( y )˜ u ] + ˜ u. Using Lemma 3.2, the interpolation in the L p spaces, ˜ u n → ˜ u in H s ( R ) , assumptions ( g ) and ( g ) we cansee that ξ n → ξ in L p ( R ) ∀ p ∈ [2 , ∞ ) , and that there exists C > such that | ξ n | ∞ ≤ C ∀ n ∈ N . Consequently, ˜ u n ( x ) = ( K ∗ ξ n )( x ) = R R K ( x − z ) ξ n ( z ) dz , where K is the Bessel kernel and satisfies thefollowing properties [21]: ( i ) K is positive, radially symmetric and smooth in R \ { } , ( ii ) there is C > such that K ( x ) ≤ C | x | s for any x ∈ R \ { } , ( iii ) K ∈ L r ( R ) for any r ∈ [1 , − s ) .Hence, arguing as in Lemma . in [4], we can see that ˜ u n ( x ) → as | x | → ∞ uniformly in n ∈ N . (3.18)Therefore, we can find R > such that ˜ u n ( x ) < a ∀| x | ≥ R ∀ n ∈ N , which yields u n ( x ) < a for any x ∈ R \ B R (˜ y n ) and n ∈ N .On the other hand, there exists ν ∈ N such that for any n ≥ ν , it holds R \ Λ ε n ⊂ R \ B rεn (˜ y n ) ⊂ R \ B R (˜ y n ) , which gives u n ( x ) < a ∀ x ∈ R \ Λ ε n . This last fact contradicts (3.16) and thus (3.15) is verified.Now, let u ε be a solution to (2.3). Since u ε satisfies (3.15) for any ε ∈ (0 , ˜ ε ) , it follows from the definitionof g that u ε is a solution to (2.1), and then ˆ u ε ( x ) = u ( x/ ε ) is a solution to (1.1) for any ε ∈ (0 , ˜ ε ) .Finally, we study the behavior of the maximum points of solutions to problem (2.1). Take ε n → + andconsider a sequence { u n } n ∈ N ⊂ H ε n of solutions to (2.1). We first notice that, by ( g ) , there exists γ ∈ (0 , a ) such that g ( ε n x, t ) t = f ( t ) t + t ∗ s ≤ V K t for any x ∈ R , ≤ t ≤ γ. (3.19)The same argument as before yields, for some R > , | u n | L ∞ ( R \B R (˜ y n )) < γ. (3.20)Moreover, up to extract a subsequence, we may assume that | u n | L ∞ ( B R (˜ y n )) ≥ γ. (3.21) Indeed, if (3.21) does not hold, we can see that (3.20) implies that | u n | ∞ < γ . Then, in view of hJ ′ ε n ( u n ) , u n i =0 and (3.19), we can see that k u n k ε n ≤ k u n k ε n + b [ u n ] s = Z R g ( ε n x, u n ) u n dx ≤ V K Z R u n dx which gives k u n k ε n = 0 , that is a contradiction. Hence, (3.21) holds true. In the light of (3.20) and (3.21), wecan deduce that the maximum point p n ∈ R of u n belongs to B R (˜ y n ) . Thus, p n = ˜ y n + q n for some q n ∈ B R .Recalling that the solution to (1.1) is of the form ˆ u n ( x ) := u n ( x/ ε n ) , we conclude that the maximum point η ε n of ˆ u n is given by η ε n := ε n ˜ y n + ε n q n . Since { q n } n ∈ N ⊂ B R is bounded and ε n ˜ y n → y with V ( y ) = V ,from the continuity of V we can infer that lim n →∞ V ( η ε n ) = V ( y ) = V . Next, we give a decay estimate for ˆ u n . Invoking Lemma . in [21], we know that there exists a positivefunction w such that < w ( x ) ≤ C | x | s , (3.22)and ( − ∆) s w + V a + bA ) w ≥ in R \ B R , (3.23)for some suitable R > , and A > is such that a + b [ u n ] s ≤ a + bA ∀ n ∈ N . Using ( f ) , the definition of g and (3.18), we can find R > sufficiently large such that ( − ∆) s ˜ u n + V a + bA ) ˜ u n ≤ ( − ∆) s ˜ u n + V a + b [˜ u n ] ) ˜ u n = 1 a + b [˜ u n ] s (cid:20) g ( ε n x + ε n y n , ˜ u n ) − (cid:18) V n − V (cid:19) ˜ u n (cid:21) ≤ a + b [˜ u n ] s (cid:20) g ( ε n x + ε n y n , ˜ u n ) − V u n (cid:21) ≤ in R \ B R . (3.24)Define R := max { R , R } > and we set c := inf B R w > and ˜ w n := ( d + 1) w − c ˜ u n , (3.25)where d := sup n ∈ N | ˜ u n | ∞ < ∞ . In what follows, we show that ˜ w n ≥ in R . (3.26)Firstly, we can observe that (3.23), (3.24) and (3.25) yield ˜ w n ≥ cd + w − cd > in B R , (3.27) ( − ∆) s ˜ w n + V a + bA ) ˜ w n ≥ in R \ B R . (3.28)Now, we argue by contradiction and we assume that there exists a sequence { ¯ x n,k } k ∈ N ⊂ R such that inf x ∈ R ˜ w n ( x ) = lim k →∞ ˜ w n (¯ x n,k ) < . (3.29)By (3.18), (3.22) and the definition of ˜ w n , it is clear that | ˜ w n ( x ) | → as | x | → ∞ , uniformly in n ∈ N . Thus, { ¯ x n,k } k ∈ N is bounded, and, up to subsequence, we may assume that there exists ¯ x n ∈ R such that ¯ x n,k → ¯ x n as k → ∞ . It follows from (3.29) that inf x ∈ R ˜ w n ( x ) = ˜ w n (¯ x n ) < . (3.30)From the minimality property of ¯ x n and the representation formula for the fractional Laplacian [18], we cansee 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 ≤ . (3.31) RACTIONAL CRITICAL KIRCHHOFF EQUATIONS 19
Taking into account (3.27) and (3.29) we can infer that ¯ x n ∈ R \ B R . This together with (3.30) and (3.31)implies ( − ∆) s ˜ w n (¯ x n ) + V a + bA ) ˜ w n (¯ x n ) < , which is impossible in view of (3.28). Hence, (3.26) is verified.According to (3.22) and (3.26), we obtain < ˜ u n ( x ) ≤ ˜ C | x | s ∀ n ∈ N ∀ x ∈ R , (3.32)for some constant ˜ C > . Since ˆ u n ( x ) = u n ( xε n ) = ˜ u n ( xε n − ˜ y n ) and η ε n = ε n ˜ y n + ε n q n , we can use (3.32) todeduce that < ˆ u n ( x ) = u n (cid:18) xε n (cid:19) = ˜ u n (cid:18) xε n − ˜ y n (cid:19) ≤ ˜ C | xε n − ˜ y n | s = ˜ C ε sn ε sn + | x − ε n ˜ y n | s ≤ ˜ C ε sn ε sn + | x − η ε n | s ∀ x ∈ R . This ends the proof of Theorem 1.1. (cid:3)
References [1] C. O. Alves, F. J. S. A. Corrêa and G. M. Figueiredo,
On a class of nonlocal elliptic problems with critical growth , Differ.Equ. Appl. (2010), no. 3, 409–417. 2[2] C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type ,Comput. Math. Appl. (2005), no. 1, 85–93. 2[3] C.O. Alves, J.M. do Ó and M.A.S. Souto, Local mountain-pass for a class of elliptic problems in R N involving criticalgrowth , Nonlinear Anal. (2001) 495–510. 3[4] C.O. Alves and O.H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in R N viapenalization method , Calc. Var. Partial Differential Equations (2016), no. 3, Art. 47, 19 pp. 1, 2, 17[5] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications , J. FunctionalAnalysis (1973), 349–381. 2, 3[6] V. Ambrosio, Ground states for a fractional scalar field problem with critical growth , Differential Integral Equations (2017), no. 1-2, 115–132. 1[7] V. Ambrosio, Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method , Ann.Mat. Pura Appl. (4) (2017), no. 6, 2043–2062. 1, 2[8] V. Ambrosio,
Concentration phenomena for critical fractional Schrödinger systems , Commun. Pure Appl. Anal. (2018),no. 5, 2085–2123. 1[9] V. Ambrosio and T. Isernia, A multiplicity result for a fractional Kirchhoff equation in R N with a general nonlinearity ,Commun. Contemp. Math. (2018), no. 5, 1750054, 17 pp. 2, 3[10] V. Ambrosio and T. Isernia, Concentration phenomena for a fractional Schrödinger-Kirchhoff type problem , Math. MethodsAppl. Sci. (2018), no.2, 615–645. 2, 3[11] G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a criticalnonlinearity , Nonlinear Anal. (2015), 699–714. 2[12] S. Bernstein,
Sur une classe d’équations fonctionnelles aux dérivées partielles , Bull. Acad. Sci. URSS. Sér. Math. [IzvestiaAkad. Nauk SSSR] (1940), 17–26. 2[13] C. Chen, Y. Kuo, and T. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions ,J. Differential Equations 250 (2011), no. 4, 1876–1908. 2[14] M. Chipot and B. Lovat,
Some remarks on nonlocal elliptic and parabolic problems , Nonlinear Anal. (1997), no. 7,4619–4627. 2[15] A. Cotsiolis and N.K. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives , J. Math.Anal. Appl. (2004), no. 1, 225–236. 6[16] J. Dávila, M. del Pino, and J. Wei,
Concentrating standing waves for the fractional nonlinear Schrödinger equation , J.Differential Equations (2014), no. 2, 858–892. 1, 2[17] M. Del Pino and P.L. Felmer,
Local mountain passes for semilinear elliptic problems in unbounded domains , Calc. Var.Partial Differential Equations, (1996), 121–137. 1, 3, 4[18] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces , Bull. Sci. math. (2012),521–573. 1, 3, 4, 18 [19] S. Dipierro, M. Medina and E. Valdinoci,
Fractional elliptic problems with critical growth in the whole of R n , Appunti.Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15.Edizioni della Normale, Pisa, 2017. viii+152 pp. 1[20] M. M. Fall, F. Mahmoudi and E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödingerequation , Nonlinearity (2015), no. 6, 1937–1961. 1, 2[21] P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian ,Proc. Roy. Soc. Edinburgh Sect. A (2012), 1237–1262. 1, 4, 17, 18[22] G. M. Figueiredo and M. Furtado,
Positive solutions for a quasilinear Schrödinger equation with critical growth , J. Dynam.Differential Equations (2012), no. 1, 13–28. 3[23] G. M. Figueiredo, G. Molica Bisci and R. Servadei, On a fractional Kirchhoff-type equation via Krasnoselskii’s genus ,Asymptot. Anal. (2015), no. 3-4, 347–361. 2[24] G.M. Figueiredo and J.R. Santos, Multiplicity and concentration behavior of positive solutions for a Schrödinger-Kirchhofftype problem via penalization method , ESAIM Control Optim. Calc. Var. (2014), no. 2, 389–415. 2[25] A. Fiscella and P. Pucci, p -fractional Kirchhoff equations involving critical nonlinearities , Nonlinear Anal. Real WorldAppl. (2017), 350–378. 2[26] A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator , Nonlinear Anal. (2014),156–170. 2[27] Y. He, G. Li and S. Peng, Concentrating bound states for Kirchhoff type problems in R involving critical Sobolev exponents ,Adv. Nonlinear Stud. (2014), no. 2, 483–510. 2[28] X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in R , J. Differ.Equ. (2012), 1813–1834. 2[29] X. He and W. Zou, Multiplicity of concentrating solutions for a class of fractional Kirchhoff equation , Manuscripta Math. (2019), no. 1-2, 159–203. 3[30] T. Isernia,
Positive solution for nonhomogeneous sublinear fractional equations in R N , Complex Var. Elliptic Equ. (2018), no. 5, 689–714. 1[31] G. Kirchhoff, Mechanik , Teubner, Leipzig, 1883. 2[32] N. Laskin,
Fractional quantum mechanics and Lévy path integrals , Phys. Lett. A (2000), no. 4-6, 298–305. 1[33] J.L. Lions,
On some questions in boundary value problems of mathematical physics , Contemporary developments in con-tinuum mechanics and partial differential equations (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Riode Janeiro, 1977), pp. 284–346, North-Holland Math. Stud., 30, North-Holland, Amsterdam-New York, 1978. 2[34] Z. Liu, M. Squassina and J. Zhang,
Ground states for fractional Kirchhoff equations with critical nonlinearity in lowdimension , NoDEA Nonlinear Differential Equations Appl. (2017), no. 4, Art. 50, 32 pp. 2, 3, 6[35] X. Mingqi, V. D. Rădulescu and B. Zhang, Combined effects for fractional Schrödinger-Kirchhoff systems with criticalnonlinearities , ESAIM: cocv, (2018), no. 3, 1249 –1273. 2[36] G. Molica Bisci, V. Rădulescu and R. Servadei, Variational methods for nonlocal fractional problems , with a forewordby Jean Mawhin. Encyclopedia of Mathematics and its Applications, 162. Cambridge University Press, Cambridge, 2016.xvi+383 pp. 1, 2, 3[37] J. Moser,
A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations , Comm.Pure Appl. Math. (1960), 457–468. 3, 14[38] N. Nyamoradi, Existence of three solutions for Kirchhoff nonlocal operators of elliptic type , Math. Commun. (2013),no. 2, 489–502. 2[39] G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for frac-tional Sobolev spaces , Calc. Var. Partial Differential Equations (2014), no. 3-4, 799–829. 4[40] S.I. Pohožaev, A certain class of quasilinear hyperbolic equations , Mat. Sb. (1975), 152–166. 2[41] P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in R N involving nonlocal operators , Rev. Mat. Iberoam. (2016), no. 1, 1–22. 2[42] P. Pucci, M. Xiang and B. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involvingthe fractional p -Laplacian in R N , Calc. Var. Partial Differential Equations, (2015), 2785–2806. 2[43] P. Rabinowitz, On a class of nonlinear Schrödinger equations
Z. Angew. Math. Phys. (1992), no. 2, 270–291. 2[44] S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in R N , J. Math. Phys. (2013), 031501.1[45] R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian , Trans. Amer. Math. Soc. (2015),no. 1, 67–102. 6[46] X. Shang, J. Zhang and Y. Yang,
On fractional Schrödinger equation in R N with critical growth , J. Math. Phys. (2013),no. 12, 121502, 20 pp. 1, 2[47] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator , Comm. Pure Appl. Math., (2007), no. 1, 67–112. 10[48] J. Wang, L. Tian, J. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problemwith critical growth , J. Differential Equations (2012), no. 7, 2314–2351. 2[49] M. Willem.
Minimax theorems , Birkhäuser, 1996. 5, 10
RACTIONAL CRITICAL KIRCHHOFF EQUATIONS 21
Vincenzo AmbrosioDipartimento di Ingegneria Industriale e Scienze MatematicheUniversità Politecnica delle MarcheVia Brecce Bianche, 1260131 Ancona (Italy)
E-mail address ::