On the Schrödinger-Born-Infeld system
aa r X i v : . [ m a t h . A P ] O c t ON THE SCHR ¨ODINGER-BORN-INFELD SYSTEM
ANTONIO AZZOLLINI, ALESSIO POMPONIO, AND GAETANO SICILIANOA
BSTRACT . In this paper we study a system which we propose as a model to de-scribe the interaction between matter and electromagnetic field from a dualistic pointof view. This system arises from a suitable coupling of the Schr ¨odinger and the Born-Infeld lagrangians, this latter replacing the role that, classically, is played by the Maxwelllagrangian.We use a variational approach to find an electrostatic radial ground state solution bymeans of suitable estimates on the functional of the action.
1. I
NTRODUCTION
In the recent years, several models have been proposed to provide a mathemati-cal description of the interaction between a charged particle and the electromagneticfield generated by itself. According to two different philosophycal concepts, the wayto perform a mathematical formulation can follow two different and, in some way,antithetical approaches.The theory developed by Born and Infeld (see [7] and [8]) introduced the idea thatboth the matter and the electromagnetic field were expression of a unique physicalentity. According to this unitarian point of view, the system giving a complete de-scription of the dynamics arose variationally starting from a nonlinear version of theMaxwell lagrangian. This unitarian approach was also taken up by Benci and Fortu-nato in [4] (see also [1] and [12]).On the other hand there is the dualistic point of view, based on the idea that the dy-namics can be described coupling equations related with particles and equations re-lated with the electromagnetic field through a suitable combination of the lagrangians.Starting from the results obtained by Benci and Fortunato [2], the literature is rich ofpapers studying models based on this latter point of view.In the past, the duality matter-electromagnetic field was usually carried out by meansof either Schr ¨odinger or Klein-Gordon lagrangian to provide the mathematical de-scription of the particle, and of the Maxwell lagrangian, or higher order approxima-tions (in the sense of Taylor series) of the Born-Infeld lagrangian (see for example [13]and [9]) to represent the electromagnetic field.Recently, Yu has proposed in [16] a dualistic model obtained coupling Klein-Gordonand Born-Infeld lagrangians and has studied the electrostatic case expressed by the
Mathematics Subject Classification.
Key words and phrases.
Schr ¨odinger-Born-Infeld equation, nonlinear electromagnetic theory.A. Azzollini and A. Pomponio are partially supported by a grant of the group GNAMPA of INDAM.A. Pomponio is partially supported also by FRA2016 of Politecnico di Bari. G. Siciliano is supportedby Capes, CNPq and Fapesp, Brazil. following system(
KGBI ) − ∆ u + ( m − ( ω + φ ) ) u = | u | p − u in R , div ∇ φ p − |∇ φ | ! = u ( ω + φ ) in R ,u ( x ) → , φ ( x ) → , as x → ∞ . As a consequence of the form of the differential operator in the second equation,a variational approach to the problem can not be performed in the usual functionalspaces. In particular, the quantity / p − |∇ φ ( x ) | makes sense when x ∈ R is suchthat |∇ φ ( x ) | < , being this inequality a necessary constraint to be considered in thefunctional setting.Inspired by [16], our aim is to propose and study a new model which represents avariant of the well-known Schr ¨odinger-Maxwell system as it was introduced in [10].Indeed we replace the usual Maxwell lagrangian with the Born-Infeld one and welook for the electrostatic solutions. The system in this case becomes( SBI ) − ∆ u + u + φu = | u | p − u in R , − div ∇ φ p − |∇ φ | ! = u in R ,u ( x ) → , φ ( x ) → , as x → ∞ , and we will refer to it as Schr ¨odinger-Born-Infeld system.At least formally, the system ( SBI ) comes variationally from the action functional F defined by F ( u, φ ) = 12 Z R (cid:0) |∇ u | + u (cid:1) + 12 Z R φu − p + 1 Z R | u | p +1 − Z R (cid:16) − p − |∇ φ | (cid:17) . Dealing with this functional presents evident difficulties for several reasons, start-ing with the definition of the functional setting. Indeed we observe that, being onthe one hand natural to consider u ∈ H ( R ) , on the other the presence of the term R R (cid:16) − p − |∇ φ | (cid:17) forces us to restrict the setting of admissible functions φ .We define(1) X := D , ( R ) ∩ { φ ∈ C , ( R ) : k∇ φ k ∞ ≤ } where D , ( R ) is the completion of C ∞ c ( R ) with respect to the norm k∇·k . Hereafterwe denote by k · k q the norm in L q ( R ) , for q ∈ [1 , + ∞ ] .We are looking for weak solutions in the following sense. Definition 1.1. A weak solution of ( SBI ) is a couple ( u, φ ) ∈ H ( R ) × X such that forall ( v, ψ ) ∈ C ∞ c ( R ) × C ∞ c ( R ) , we have Z R ∇ u · ∇ v + uv + φuv = Z R | u | p − uv Z R ∇ φ · ∇ ψ p − |∇ φ | = Z R u ψ. Observe that the boundary condition at infinity is encoded in the functional space.Of course, the fact that the setting H ( R ) × X is not a vector space is a nontrivialobstacle to our variational approach. In particular, to compute variations with respect N THE SCHR ¨ODINGER-BORN-INFELD SYSTEM 3 to φ along the direction established by a generic smooth and compactly supportedfunction, we need to require in advance that k∇ φ k ∞ < . This fact brings with it aconcrete complication, for example in dealing with the reduction method which is astandard tool used in this kind of problems (see, for example, [2, 3, 16]). Indeed, thestrongly indefinite nature of the functional can be classically removed showing that,for any u ∈ H ( R ) fixed, there exists a unique φ u ∈ X solution of the second equationof system ( SBI ) and reducing the problem to that of finding critical points of the (nomore strongly indefinite) one-variable functional I ( u ) = F ( u, φ u ) , defined on H ( R ) (see Section 2 for more details).As a consequence, we are led to consider a preliminary minimizing problem on theset X and then, because of the bad properties of X itself, we have to study the relationbetween solutions of this minimizing problem and solutions of the second equation(with respect to φ , being u fixed). This second step is one of the questions left openfor ( KGBI ) in [16], which has been recently solved in [6] in a radial setting. For thisreason, and also in order to overcome difficulties related with compactness, we willrestrict our study to radial solutions. So, let us introduce our functional framework:we set H r ( R ) = { u ∈ H ( R ) | u is radially symmetric } and X r = { φ ∈ X | φ is radially symmetric } . Our main results are the following
Theorem 1.2.
For any p ∈ (5 / , , the problem ( SBI ) possesses a radial ground statesolution, namely a solution ( u, φ ) ∈ H r ( R ) × X r minimizing the functional F among all thenontrivial radial solutions. Moreover both u and φ are of class C ( R ) . What immediately stands out is the unusual range where p varies. It follows fromthe fact that, in view of the application of the mountain pass theorem, we need to finda point with a sufficiently large norm where the functional is negative. In order todo this, usually one computes the reduced one-variable functional I on curves of thetype t ∈ (0 , + ∞ ) u t := t α u ( t β · ) ∈ H r ( R ) , and look for suitable values of α and β for which I ( u t ) < for large values of t . How-ever, in our case, because of the lack of homogeneity and since a precise expression of φ u t is not available, we need to proceed by means of estimates of φ u which lead, as aconsequence, to lose something in terms of powers p .Summing up, our aim in this paper is to propose the new model problem ( SBI ) andgive a positive answer concerning the existence of solutions, at least for p ∈ (5 / , .We leave as an open problem the case of smaller p and the existence of non-radialsolutions.The paper is organized as follows: in Section 2 we introduce the functional settingand present some preliminary results, while in Section 3 we prove Theorem 1.2.We finish this section with some notations. In the following we denote by k · k thenorm in H ( R ) and by c, c i , C, C i arbitrary fixed positive constants which can varyfrom line to line.2. F UNCTIONAL SETTING AND PRELIMINARY RESULTS
We start recalling some properties of the ambient space X defined in (1). Lemma 2.1 (Lemma 2.1 of [6]) . The following assertions hold:
A. AZZOLLINI, A. POMPONIO, AND G. SICILIANO (i) X is continuously embedded in W ,p ( R ) , for all p ∈ [6 , + ∞ ) ;(ii) X is continuously embedded in L ∞ ( R ) ;(iii) if φ ∈ X , then lim | x |→∞ φ ( x ) = 0 ;(iv) X is weakly closed;(v) if ( φ n ) n ⊂ X is bounded, there exists ¯ φ ∈ X such that, up to a subsequence, φ n ⇀ ¯ φ weakly in X and uniformly on compact sets. As already observed in the Introduction, the functional F is strongly indefinite on H ( R ) × X from above and from below, and so we will consider a reduced one-variable functional, solving the second equation of ( SBI ), for any fixed u ∈ H r ( R ) .Let us start considering the functional E : H ( R ) × X → R defined as E ( u, φ ) = Z R (cid:16) − p − |∇ φ | (cid:17) − Z R φu . The following lemma holds.
Lemma 2.2.
For any u ∈ H ( R ) fixed, there exists a unique φ u ∈ X such that the followingproperties hold:(i) φ u is the unique minimizer of the functional E ( u, · ) : X → R and E ( u, φ u ) ,namely (2) Z R φ u u > Z R (cid:16) − p − |∇ φ u | (cid:17) ; (ii) φ u > and φ u = 0 if and only if u = 0 ;(iii) if φ is a weak solution of the second equation of system ( SBI ) , then φ = φ u and itsatisfies the following equality (3) Z R |∇ φ u | p − |∇ φ u | = Z R φ u u . Moreover, if u ∈ H r ( R ) , then φ u ∈ X r is the unique weak solution of the second equation ofsystem ( SBI ) .Proof. Points (i), (ii) and (iii) are an immediate consequence of Theorems 1.3 andLemma 2.12 of [6]. For the second part of the statement we refer to [6, Theorem1.4]. (cid:3)
Remark 2.3.
We point out that, as stated in [6, Remark 5.5], if w n → w in L p ( R ) , with p ∈ [1 , + ∞ ) then φ w n → φ w in L ∞ ( R ) . By Lemma 2.2, we can deal with the following one-variable functional defined on H ( R ) as I ( u ) = F ( u, φ u )= 12 Z R (cid:0) |∇ u | + u (cid:1) + 12 Z R φ u u − p + 1 Z R | u | p +1 − Z R (cid:16) − p − |∇ φ u | (cid:17) = 12 Z R (cid:0) |∇ u | + u (cid:1) − p + 1 Z R | u | p +1 − E ( u, φ u ) . Proposition 2.4.
The functional I is of class C and for every u, v ∈ H ( R ) , I ′ ( u )[ v ] = Z R ∇ u · ∇ v + Z R uv + Z R φ u uv − Z R | u | p − uv. N THE SCHR ¨ODINGER-BORN-INFELD SYSTEM 5
Proof.
Arguing as in [16], let us show that I ( u + v ) − I ( u ) − DI ( u )[ v ] = o ( v ) , as v → , where DI ( u )[ v ] := Z R ∇ u · ∇ v + Z R uv + Z R φ u uv − Z R | u | p − uv which is trivially linear and continuous in v .We set I ( u + v ) − I ( u ) − DI ( u )[ v ] = A + A + A where A := 12 Z R |∇ ( u + v ) | − Z R |∇ u | − Z R ∇ u · ∇ v,A := − p + 1 Z R | u + v | p +1 + 1 p + 1 Z R | u | p +1 + Z R | u | p − uv,A := 12 Z R v − E ( u + v, φ u + v ) + 12 E ( u, φ u ) − Z R φ u uv. Clearly A = o ( v ) , A = o ( v ) . Now observe that by point (i) of Lemma 2.2 we have E ( u + v, φ u ) > E ( u + v, φ u + v ) , sothat an explicit computation gives A > Z R v − E ( u + v, φ u ) + 12 E ( u, φ u ) − Z R φ u uv = 12 Z R v + 12 Z φ u v > Z R φ u v = o ( v ) being (cid:12)(cid:12)(cid:12) Z R v φ u (cid:12)(cid:12)(cid:12) C k v k k φ u k ∞ . Analogously, once again by point (i) of Lemma 2.2,being E ( u, φ u ) E ( u, φ u + v ) , we get A Z R v − E ( u + v, φ u + v ) + 12 E ( u, φ u + v ) − Z R φ u uv = 12 Z R v + Z R φ u + v uv + 12 Z R φ u + v v − Z R φ u uv = o ( v ) + 12 Z φ u + v v + Z R ( φ u + v − φ u ) uv o ( v ) + C k v k k φ u + v k ∞ + C k u kk v kk φ u + v − φ u k ∞ = o ( v ) , in view of Remark 2.3. Hence A = o ( v ) and the differentiability of I is proved.Finally, let us prove the continuity of the map u ∈ H ( R ) φ u u ∈ L ( H ( R ); R ) , from which we easily deduce the continuity of DI : H ( R ) → L ( H ( R ); R ) . Let u n → u in H ( R ) . Observe that uniformly in v ∈ H ( R ) , with k v k , Z R (cid:12)(cid:12)(cid:12) φ u n u n − φ u u (cid:12)(cid:12)(cid:12) | v | Z R | φ u n || u n − u || v | + Z R | φ u n − φ u || u || v | = o n (1) , again by Remark 2.3. The conclusion follows. (cid:3) A. AZZOLLINI, A. POMPONIO, AND G. SICILIANO
Proposition 2.5. If ( u, φ ) ∈ H ( R ) ×X is a weak nontrivial solution of ( SBI ) , then φ = φ u and u is a critical point of I . On the other hand, if u ∈ H r ( R ) \ { } is a critical point of I ,then ( u, φ u ) is a weak nontrivial solution of ( SBI ) .Proof. The first part of the statement is a consequence of [6, Proposition 2.6] andProposition 2.4, while the second part follows by Lemma 2.2 and Proposition 2.4. (cid:3)
In the next proposition we are going to prove that H r ( R ) is a natural constraint forthe functional I . Proposition 2.6. If u ∈ H r ( R ) is a critical point of I | H r ( R ) , then u is a critical point of I .Proof. Denote by O (3) the group of rotations in R and for any g ∈ O (3) consider theaction induced on H ( R ) , that is T g : u ∈ H ( R ) u ◦ g ∈ H ( R ) . Clearly H r ( R ) is the set of the fixed points for the group T = { T g } g ∈ O (3) namely H r ( R ) = { u ∈ H ( R ) | T g u = u for all g ∈ O (3) } . Then the conclusion can be achieved by the Palais’ Principle of Symmetric Criticality,if we show that I is invariant under the action of T , that is I ( T g u ) = I ( u ) , for all g ∈ O (3) , u ∈ H ( R ) . Actually it is sufficient to show that φ T g u = T g φ u for any u ∈ H ( R ) and for all g ∈ O (3) . To this aim, by Lemma 2.2, we have E ( u, T g − φ T g u ) = E ( T g u, φ T g u ) E ( T g u, T g φ u ) = E ( u, φ u ) and so, by the uniqueness of the minimizer of E ( u, · ) , we conclude that φ u = T g − φ T g u as desired. (cid:3) The following technical lemma will be useful to study the geometry of the func-tional I . Lemma 2.7.
Let q be in [2 , . Then there exist positive constants C and C ′ such that, for any u ∈ H ( R ) , we have k∇ φ u k q − q C k u k q ∗ ) ′ C ′ k u k , where q ∗ is the critical Sobolev exponent related to q and ( q ∗ ) ′ is its conjugate exponent,namely q ∗ = 3 q − q and ( q ∗ ) ′ = 3 q q − . Proof.
Since k∇ φ u k ∞ and q < , k φ u k q ∗ C k∇ φ u k q = C (cid:18)Z R |∇ φ u | |∇ φ u | q − (cid:19) q C k∇ φ u k q , so, by (2) and being q ∗ ) ′ ∈ [2 , , we have k∇ φ u k C Z R (cid:16) − p − |∇ φ u | (cid:17) C Z R φ u u C k φ u k q ∗ k u k q ∗ ) ′ C k∇ φ u k q k u k q ∗ ) ′ and we get the conclusion. (cid:3) We conclude this section showing that the radial weak solutions of (
SBI ) are actu-ally classical and satisfy a Pohozaev type identity.
N THE SCHR ¨ODINGER-BORN-INFELD SYSTEM 7
Proposition 2.8. If ( u, φ ) ∈ H r ( R ) × X r is a weak solution of ( SBI ) , then both u and φ are of class C ( R ) .Proof. Since u ∈ H r ( R ) , by [6, Theorem 3.2] we deduce that φ ∈ C ( R ) . Looking atthe first equation in the system and by using a bootstrap argument, we conclude that u ∈ C ( R ) . We define ϕ : [0 , + ∞ [ → R such that for any r > ϕ ( r ) = φ ( | x | ) where x ∈ R is arbitrarily chosen in such a way that | x | = r .From now on, we proceed as in [5, Lemma 1, page 329]. Since φ is radial and satisfiesthe second equation in a weak sense, we deduce that D ϕ ′ r p − | ϕ ′ | ! = − u r , in (0 , + ∞ ) where the symbol D denotes the derivative in the sense of distributions.Since on the right hand side we have a continuous function, the derivative actuallyhas to be meant in the classical sense. So, integrating in (0 , r ) and since ϕ ′ (0) = 0 ,(4) ϕ ′ ( r ) p − | ϕ ′ ( r ) | = − r Z r u ( s ) s ds =: f ( r ) ∈ C (cid:0) (0 , + ∞ ) (cid:1) . On the one hand, by (4), we deduce that, for r > , we have f ′ ( r ) = 2 r Z r u ( s ) s ds − u ( r ) and then lim r → f ′ ( r ) = − u (0) .On the other hand, again by (4), lim r → f ( r ) r = lim r → − r Z r u ( s ) s ds = − u (0) . We conclude that there exists f ′ (0) and lim r → f ′ ( r ) = f ′ (0) . Then f ∈ C (cid:0) [0 , + ∞ ) (cid:1) .By some computations, by (4), we have ϕ ′ ( r ) = f ( r ) p f ( r ) ∈ C (cid:0) [0 , + ∞ ) (cid:1) and we are done. (cid:3) Proposition 2.9. If ( u, φ ) is a solution of ( SBI ) of class C ( R ) , then the following Pohozaevtype identity is satisfied: (5) Z R |∇ u | + 32 Z R u + 2 Z R |∇ φ | p − |∇ φ | − Z R (cid:16) − p − |∇ φ | (cid:17) = 3 p + 1 Z R | u | p +1 . A. AZZOLLINI, A. POMPONIO, AND G. SICILIANO
Proof.
Arguing as in [11], for every
R > , we have Z B R − ∆ u ( x · ∇ u ) = − Z B R |∇ u | − R Z ∂B R | x · ∇ u | + R Z ∂B R |∇ u | , (6) Z B R u ( x · ∇ u ) = − Z B R u + R Z ∂B R u , (7) Z B R φu ( x · ∇ u ) = − Z B R u ( x · ∇ φ ) − Z B R φu + R Z ∂B R φu , (8) Z B R | u | p − u ( x · ∇ u ) = − p + 1 Z B R | u | p +1 + Rp + 1 Z ∂B R | u | p +1 , (9)where B R is the ball of R centered in the origin and with radius R .Moreover, denoting by δ ij the Kronecker symbols, since for any i, j = 1 , , , Z B R ∂ i ∂ i φ p − |∇ φ | ! x j ∂ j φ = − Z B R ∂ i φ ∂ j φ p − |∇ φ | δ ij − Z B R ∂ i φ ∂ i,j φ p − |∇ φ | x j + Z ∂B R ∂ i φ ∂ j φ p − |∇ φ | x i x j | x | , we have Z B R − div ∇ φ p − |∇ φ | ! ( x · ∇ φ ) = − X i,j =1 Z B R ∂ i ∂ i φ p − |∇ φ | ! x j ∂ j φ = Z B R |∇ φ | p − |∇ φ | + X j =1 Z B R ∂ j (cid:16) − p − |∇ φ | (cid:17) x j − X i,j =1 Z ∂B R ∂ i φ ∂ j φ p − |∇ φ | x i x j | x | = Z B R |∇ φ | p − |∇ φ | − Z B R (cid:16) − p − |∇ φ | (cid:17) + R Z ∂B R (cid:16) − p − |∇ φ | (cid:17) − X i,j =1 Z ∂B R ∂ i φ ∂ j φ p − |∇ φ | x i x j | x | . (10)Multiplying the first equation of ( SBI ) by x · ∇ u and the second equation by x · ∇ φ and integrating on B R , by (6), (7), (8), (9) and (10) we get, respectively, − Z B R |∇ u | − R Z ∂B R | x · ∇ u | + R Z ∂B R |∇ u | − Z B R u + R Z ∂B R u − Z B R u ( x · ∇ φ ) − Z B R φu + R Z ∂B R φu = − p + 1 Z B R | u | p +1 + Rp + 1 Z ∂B R | u | p +1 , (11) N THE SCHR ¨ODINGER-BORN-INFELD SYSTEM 9 and Z B R u ( x · ∇ φ ) = Z B R |∇ φ | p − |∇ φ | − Z B R (cid:16) − p − |∇ φ | (cid:17) + R Z ∂B R (cid:16) − p − |∇ φ | (cid:17) − X i,j =1 Z ∂B R ∂ i φ ∂ j φ p − |∇ φ | x i x j | x | . (12)Substituting (12) into (11), since all the boundary integrals go to zero as R → + ∞ (wecan repeat the arguments of [5]), by (3) we get the conclusion. (cid:3)
3. P
ROOFS OF THE MAIN RESULTS
Using an idea from [14, 15], we look for bounded Palais-Smale sequences of thefollowing perturbed functionals I λ ( u ) = 12 Z R ( |∇ u | + u ) + 12 Z R φ u u − Z R (cid:16) − p − |∇ φ u | (cid:17) − λp + 1 Z R | u | p +1 , for almost all λ near . Then we will deduce the existence of a non-trivial criticalpoint v λ of the functional I λ at the mountain pass level. Afterward, we study theconvergence of the sequence ( v λ ) λ , as λ goes to 1 (observe that I = I ).We begin applying a slightly modified version of the monotonicity trick due to [14,15]. Proposition 3.1.
Let (cid:0) X, k · k (cid:1) be a Banach space and J ⊂ R + an interval. Consider a familyof C functionals I λ on X defined by I λ ( u ) = A ( u ) − λB ( u ) , for λ ∈ J, with B non-negative and either A ( u ) → + ∞ or B ( u ) → + ∞ as k u k → + ∞ and such that I λ (0) = 0 . For any λ ∈ J , we set Γ λ := { γ ∈ C ([0 , , X ) | γ (0) = 0 , I λ ( γ (1)) < } . Assume that for every λ ∈ J , the set Γ λ is non-empty and c λ := inf γ ∈ Γ λ max t ∈ [0 , I λ ( γ ( t )) > . Then for almost every λ ∈ J , there is a sequence ( v n ) n ⊂ X such that (i) ( v n ) n is bounded in X ; (ii) I λ ( v n ) → c λ , as n → + ∞ ; (iii) I ′ λ ( v n ) → in the dual space X − of X , as n → + ∞ . In our case X = H r ( R ) A ( u ) = 12 Z R ( |∇ u | + u ) + 12 Z R φ u u − Z R (cid:16) − p − |∇ φ u | (cid:17) ,B ( u ) = 1 p + 1 Z R | u | p +1 . Observe that, by (2), A ( u ) → + ∞ as k u k → + ∞ . Proposition 3.2.
For all λ ∈ [1 / , , the set Γ λ is not empty. Proof.
Fix λ ∈ [1 / , and u ∈ H r ( R ) \ { } , then, by Lemma 2.7 and for q ∈ [2 , , wehave I λ ( u ) k u k + 12 Z R φ u u − λp + 1 k u k p +1 p +1 k u k + c k φ u k k u k − λp + 1 k u k p +1 p +1 k u k + c k∇ φ u k k u k − λp + 1 k u k p +1 p +1 k u k + c k u k q − q − − λp + 1 k u k p +1 p +1 . Therefore, if λ ∈ [1 / , and u ∈ H r ( R ) \ { } and t > , we infer that I λ ( tu ) c t + c t q − q − − c λt p +1 . Since p ∈ (5 / , , we can find q ∈ [2 , such that I λ ( tu ) < , for t sufficiently large. (cid:3) Proposition 3.3.
For any λ ∈ [1 / , , there exist α > and ρ > , sufficiently small, suchthat I λ ( u ) > α , for all u ∈ H ( R ) , with k u k = ρ . As a consequence c λ > α .Proof. The conclusion follows easily by Lemma 2.2. (cid:3)
Proposition 3.4.
For almost every λ ∈ J , there exists u λ ∈ H r ( R ) , u λ = 0 , such that I ′ λ ( u λ ) = 0 and I λ ( u λ ) = c λ .Proof. By Propositions 3.2 and 3.3 we can apply the monotonicity trick (Proposition 3.1)and we argue that, for almost every λ ∈ J there exists a bounded Palais-Smale se-quence ( u n ) n ⊂ H r ( R ) for the functional I λ at level c λ , namely as n → + ∞ , I λ ( u n ) → c λ , I ′ λ ( u n ) → . Fix such a λ ∈ J . Exploiting compactness results holding for H r ( R ) , we have thatthere exists u λ ∈ H r ( R ) such that, up to subsequences, u n ⇀ u λ weakly in H r ( R ) , (13) u n → u λ in L s ( R ) , < s < ,u n → u λ a.e. in R . By [6, Remark 5.5], we infer that φ n := φ u n → φ u λ =: φ λ , weakly in D , ( R ) (anduniformly in R ) so we conclude that, for every v ∈ H r ( R ) , lim n I ′ λ ( u n )[ v ] = I ′ λ ( u λ )[ v ] = 0 that is u λ is a critical point of I λ .Moreover, since the following convergence holds(14) (cid:12)(cid:12)(cid:12)(cid:12)Z R φ n u n − Z R φ λ u λ (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)Z R φ n ( u n − u λ ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z R ( φ n − φ λ ) u λ (cid:12)(cid:12)(cid:12)(cid:12) −−−−→ n → + ∞ , taking into account that I ′ λ ( u n )[ u n ] = o n (1) and I ′ λ ( u λ )[ u λ ] = 0 , by Proposition 2.4 itfollows lim n k u n k = lim n (cid:18)Z R | u n | p +1 − Z R φ u n u n (cid:19) = Z R | u λ | p +1 − Z R φ λ u λ = k u λ k . By this and (13) we deduce that u n → u λ in H r ( R ) and then, by (14), < c λ = lim n I λ ( u n ) = I λ ( u λ ) N THE SCHR ¨ODINGER-BORN-INFELD SYSTEM 11 which concludes the proof. (cid:3)
Now we are ready to prove our main result.
Proof of Theorem 1.2.
By Proposition 3.4, there exists a sequence ( λ n ) n ⊂ J such that λ n ր and, for all n ∈ N , there exists u n ∈ H r ( R ) \ { } such that I λ n ( u n ) = c λ n , (15) I ′ λ n ( u n ) = 0 in ( H ( R )) ′ . For the sake of brevity, we will denote φ n := φ u n . By (3), (5) and since I ′ λ n ( u n )[ u n ] = 0 ,we have Z R |∇ u n | + 32 Z R u n + 2 Z R |∇ φ n | p − |∇ φ n | − Z R (cid:16) − p − |∇ φ n | (cid:17) = 3 λ n p + 1 Z R | u n | p +1 Z R |∇ u n | + Z R u n + Z R |∇ φ n | p − |∇ φ n | = λ n Z R | u n | p +1 . Multiplying the first equation by α/ and the second one by β/ ( p + 1) and summing,we have ( α + β ) λ n p + 1 Z R | u n | p +1 = (cid:18) α βp + 1 (cid:19) Z R |∇ u n | + (cid:18) α βp + 1 (cid:19) Z R u n + (cid:18) α βp + 1 (cid:19) Z R |∇ φ n | p − |∇ φ n | − α Z R (cid:16) − p − |∇ φ n | (cid:17) . Assuming, in particular, α = 1 − β , we get(16) λ n p + 1 Z R | u n | p +1 = (cid:18)
16 + β (5 − p )6( p + 1) (cid:19) Z R |∇ u n | + (cid:18)
12 + β (1 − p )2( p + 1) (cid:19) Z R u n + (cid:18)
23 + β (1 − p )3( p + 1) (cid:19) Z R |∇ φ n | p − |∇ φ n | − (cid:18) − β (cid:19) Z R (cid:16) − p − |∇ φ n | (cid:17) . Therefore, since for all t ∈ [0 , − √ − t t √ − t , substituting (16) into (15), we have c λ n = I λ n ( u n ) = (cid:18) − β (5 − p )6( p + 1) (cid:19) Z R |∇ u n | + β ( p − p + 1) Z R u n + (cid:18) β (2 p − p + 1) − (cid:19) Z R |∇ φ n | p − |∇ φ n | − β Z R (cid:16) − p − |∇ φ n | (cid:17) > (cid:18) − β (5 − p )6( p + 1) (cid:19) Z R |∇ u n | + β ( p − p + 1) Z R u n + (cid:18) β (2 p − p + 1) − − β (cid:19) Z R |∇ φ n | p − |∇ φ n | . Since p > , there exists a constant β such that all the coefficients in the previousinequality are positive and so, by the boundedness of ( c λ n ) n (indeed the map λ c λ is non-increasing), we infer the boundedness of the sequence ( u n ) n in H r ( R ) , too. Now, arguing similarly as in the proof of Proposition 3.4, we can easily prove theexistence of a nontrivial critical point u of I . Hence we have S r := (cid:8) u ∈ H r ( R ) \ { } | I ′ ( u ) = 0 (cid:9) = ∅ . Moreover, any u ∈ S r satisfies k u k Z R |∇ u | + Z R u + Z R |∇ φ u | p − |∇ φ u | = Z R | u | p +1 C k u k p +1 , and therefore inf u ∈S r k u k > . Since we have that I ( u ) > c k u k for all u ∈ S r , we conclude that σ r := inf u ∈S r I ( u ) > . Let ( u n ) n ⊂ S r such that I ( u n ) → σ r . Arguing as before we have that the sequence isbounded. Finally, as in the proof of Proposition 3.4, there exists u ∈ H r ( R ) criticalpoint of I such that, up to subsequences, u n → u in H r ( R ) . Then ( u, φ u ) is a radialground state solution by Proposition 2.5.Finally, by Proposition 2.8 we conclude that u and φ u are of class C ( R ) . (cid:3) R EFERENCES [1] A. Azzollini, V. Benci, T. D’Aprile, D. Fortunato,
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