Ground states of some coupled nonlocal fractional dispersive PDEs
aa r X i v : . [ m a t h . A P ] J a n Ground states of some coupled nonlocalfractional dispersive PDEs
Eduardo Colorado ∗ Departamento de Matem´aticas, Universidad Carlos III de MadridAvda. Universidad 30, 28911 Legan´es (Madrid), Spain.& Instituto de Ciencias Matem´aticas, ICMAT(CSIC-UAM-UC3M-UCM)C/Nicol´as Cabrera 15, 28049 Madrid, Spaine-mail: [email protected], & [email protected] to the memory of Anna Aloe
Abstract
We show the existence of ground state solutions to the following stationary sys-tem coming from some coupled fractional dispersive equations such as: nonlinearfractional Schr¨odinger (NLFS) equations (for dimension n = 1 , ,
3) or NLFSand fractional Korteweg-de Vries equations (for n = 1), (cid:26) ( − ∆) s u + λ u = u + βuv, u ∈ W s, ( R n )( − ∆) s v + λ v = v + βu , v ∈ W s, ( R n ) , where λ j > j = 1 , β ∈ R , n = 1 , ,
3, and n < s <
1. Precisely, we provethe existence of a positive radially symmetric ground state for any β > . . Key words . Nonlinear Fractional Schr¨odinger Equations, Fractional Korteweg-de Vries equations,Variational Methods, Critical Point Theory, Ground States.
In this paper we study the existence of ground state solutions to the followingstationary system coming from some coupled nonlocal fractional dispersive equa-tions such as: nonlinear fractional Schr¨odinger (NLFS) equations (for dimension ∗ Partially supported by the Ministry of Economy and Competitiveness of Spain andFEDER under Research Project MTM2016-80618-P, and by the INdAM - GNAMPA Project2017 “Teoria e modelli non locali”. = 1 , ,
3) or NLFS and fractional Korteweg-de Vries equations (FKdV) (for n = 1) (cid:26) ( − ∆) s u + λ u = u + βuv, u ∈ W s, ( R n )( − ∆) s v + λ v = v + βu , v ∈ W s, ( R n ) , (1.1)where W s, ( R n ) denotes the fractional Sobolev space, n = 1 , , λ j > j = 1 ,
2, the coupling factor β ∈ R , and the fraction n < s < ∗ s = 2 nn − s if n > s ,and 2 ∗ s = ∞ if n ≤ s . As a consequence, since n < s < ∗ s > − ∆) s , 0 < s <
1, is a nonlocaldiffusive type operator. It arises in several physical phenomena like flamespropagation and chemical reactions in liquids, population dynamics, geophysicalfluid dynamics, in probability, American options in finance, in α -stable L´evyprocesses, etc; see for instance [7, 11, 20].In the one-dimensional case, when s = 1, (1.1) comes from the followingsystem of coupled nonlinear Sch¨odinger (NLS) and Korteweg-de Vries (KdV)equations (cid:26) if t + f xx + | f | f + βf g = 0 g t + g xxx + gg x + β ( | f | ) x = 0 , (1.2)where f = f ( x, t ) ∈ C while g = g ( x, t ) ∈ R , and β ∈ R is the real couplingcoefficient. System (1.2) appears in phenomena of interactions between shortand long dispersive waves, arising in fluid mechanics, such as the interactionsof capillary - gravity water waves. Indeed, f represents the short-wave, while g stands for the long-wave. For more details, see for instance [2, 21, 29] and thereferences therein.Looking for “traveling-wave” solutions, namely solutions to (1.2) of the form( f ( x, t ) , g ( x, t )) = (cid:0) e iωt e i c x u ( x − ct ) , v ( x − ct ) (cid:1) with u, v real functions,and choosing λ = ω + c , λ = c , one finds that u, v solve the following problem (cid:26) − u ′′ + λ u = u + βuv − v ′′ + λ v = v + βu . (1.3)This system (1.3) has been studied, among others, in [2, 3, 18, 19, 23, 24]. Also,note that system (1.3) corresponds to system (1.1) when s = 1 and n = 1.On the other hand, for n = 2 ,
3, and s = 1, system (1.1) corresponds to(1.3) (cid:26) − ∆ u + λ u = u + βuv − ∆ v + λ v = v + βu , (1.4)for which the existence of bound and ground states have been studied in [18, 19].We observe that system (1.4) can be seen as a stationary version of a timedependent coupled NLS system when one looks for solitary wave solutions, and( u, v ) are the corresponding standing waves solutions of (1.4) (see for instance[19, section 6]). It is well known that systems of NLS-NLS time-dependent2quations have applications in nonlinear Optics, Hartree-Fock theory for Bose-Einstein condensates, among other physical phenomena; see for instance theearlier mathematical works [1, 4, 5, 6, 9, 33, 36, 37, 38], the more recent list (farfrom complete) [15, 17, 22, 35, 39] and references therein. See also a close relatedwork; [16], in which was studied a close system of coupled NLFS equations.Here we are interested in system (1.1), consisting of coupled NLS equationsinvolving the so called fractional Laplacian operator (or fractional Schr¨odingeroperator, ( − ∆) s + λ Id).Note that in dimension n = 1, (1.1) can also be seen as a system of coupledNLFS-FKdV equations. In this case, (1.1) is the corresponding stationary sys-tem when one looks for travelling-wave solutions of the following time-dependentsystem, (cid:26) if t − A s f + | f | f + βf g = 0 g t − ( A s g ) x + gg x + β ( | f | ) x = 0 , (1.5)where A s stands for the nonlocal fractional Laplacian ( − ∆) s in dimension n = 1.While for n = 1 , ,
3, (1.1) can be seen as the stationary system when onelooks for standing wave solutions of the following time-dependent system ofcoupled NLFS equations, (cid:26) if t − ( − ∆) s f + | f | f + βf g = 0 ig t − ( − ∆) s g + β | f | = 0 . (1.6)The main goal of this manuscript is to demonstrate that for any β > e u = ( e u, e v ) ∈ W s, ( R n ) × W s, ( R n ); see Theorems 4.1, 4.2.Notice that, for any β ∈ R , (1.1) has a unique semi-trivial positive radiallysymmetric solution, that we denote by v = (0 , V ), where V ( x ) is the uniquepositive radially symmetric ground state of − (∆) s v + λ v = v in W s, ( R n );[27, 28]. Since we are interested in positive ground states, then we have toshow that they are different from the semi-trivial solution v . To do so, we willdemonstrate some properties of the semi-trivial solution which will allow us toshow that v is not a ground state. For example, we will show that there exists aconstant Λ > β > Λ, v is a saddle point of the associated energyfunctional constrained on the corresponding Nehari Manifold, which actually isa natural restriction. When β < Λ then v is a strict local minimum of theenergy functional on the Nehari Manifold. In this case, we exclude that v isa ground state by the construction of a function in the Nehari Manifold withenergy lower than the energy of v . Precisely, we will demonstrate that thereexists a positive radially symmetric ground state of (1.1), e u = v , either: β > Λ(see Theorem 4.1) or 0 < β ≤ Λ and λ large enough (see Theorem 4.2).The paper is organized as follows. In Section 2 we introduce notation andpreliminaries, dealing with some background on the fractional Laplacian andwe give the definition of ground state. Section 3 contains some results on themethod of the natural constraint and the main properties about the semi-trivialsolution v , that we will use in the proof of the main existence results statedand proved in Section 4. Finally, in Section 5 we study the existence of groundstates for some systems with an arbitrary number of coupled equations.3 Preliminaries and Notation
The nonlocal fractional Laplacian operator ( − ∆) s in R n is defined on theSchwartz class of functions g ∈ S through the Fourier transform,[( − ∆) α g ] ∧ ( ξ ) = (2 π | ξ | ) α b g ( ξ ) , (2.1)or via the Riesz potential, see for example [31, 40]. Note that s = 1 correspondsto the standard local Laplacian operator. See also [32, 25, 27, 28], where thefractional Schr¨odinger operator (( − ∆) s +Id) is defined and are analyzed someproblems dealing with.There is another way to define this operator. If s = 1 / u in the whole space R n , can be calculated asthe normal derivative on the boundary of its harmonic extension to the upperhalf-space R n +1+ , this is so-called Dirichlet to Neumann operator. Caffarelli-Silvestre; [14], have shown that this operator can be realized in a local way byusing one more variable and the so called s -harmonic extension.More precisely, given u a regular function defined in R n we define its s -harmonic extension to the upper half-space R n +1+ by w = Ext s (u), as the solutionto the problem ( − div( y − s ∇ w ) = 0 in R n +1+ w = u on R n × { y = 0 } . (2.2)The main relevance of the s -harmonic extension comes from the following iden-tity lim y → + y − s ∂w∂y ( x, y ) = − κ s ( − ∆) s u ( x ) , (2.3)where κ s is a positive constant. The above Dirichlet-Neumann procedure (2.2)-(2.3) provides a formula for the fractional Laplacian, equivalent to that obtainedfrom Fourier Transform by (2.1). In that case, the s -harmonic extension andthe fractional Laplacian have explicit expressions in terms of the Poisson andthe Riesz kernels, respectively, w ( x, y ) = P sy ∗ u ( x ) = c n,s y s Z R n u ( z )( | x − z | + y ) n +2 s dz, ( − ∆) s u ( x ) = d n,s P.V. Z R n u ( x ) − u ( z ) | x − z | n +2 s dz. (2.4)The natural functional spaces are the homogeneous fractional Sobolev space˙ H s ( R n ) and the weighted Sobolev space X s ( R n +1+ ), that can be defined as thecompletion of C ∞ ( R n +1+ ) and C ∞ ( R n ), respectively, under the norms k φ k X s = κ s Z R n +1+ y − s |∇ φ ( x, y ) | dxdy, k ψ k H s = Z R n | πξ | s | b ψ ( ξ ) | dξ = Z R n | ( − ∆) s ψ ( x ) | dx, κ s is the constant in (2.3). Notice that, the constants in (2.4) and κ s satisfy the identity s c n,s κ s = d n,s , and their explicit value can be seen in [12]. Remark 2.1
The s -harmonic extension operator defined by (2.2) is an isometrybetween the spaces ˙ H s ( R n ) and X s ( R n +1+ ), i.e., k ϕ k ˙ H s = k E s ( ϕ ) k X s , ∀ ϕ ∈ ˙ H s ( R n ) . (2.5)Even more, we have the following inequality for the trace Tr( w ) = w ( · , k Tr( w ) k ˙ H s ≤ k w k X s , ∀ w ∈ X s ( R n +1+ ) , (2.6)see [12] for more details.Let us introduce the following notation: • E = W s, ( R n ), denotes the fractional Sobolev space, endowed with scalarproduct and norm( u | v ) j = Z R n (cid:2) ( − ∆) s u ( − ∆) s v + λ j uv (cid:3) dx, k u k j = ( u | u ) j , j = 1 , • E = E × E ; the elements in E will be denoted by u = ( u, v ); as a norm in E we will take k u k = k u k E = k u k + k v k ; • X = X s ( R n +1+ ), X = X × X ; • for u ∈ E , the notation u ≥ , resp. u > , means that u, v ≥
0, resp. u, v >
0, for all j = 1 , Remark 2.2
If we define ∂w∂ν s = − κ s lim y → + y − s ∂w∂y , we can reformulate the main problem (1.1) as − div( y − s ∇ w ) = 0 in R n +1+ − div( y − s ∇ w ) = 0 in R n +1+ ∂w ∂ν s + λ w = w + βw w on R × { y = 0 } ∂w ∂ν s + λ w = w + βw on R × { y = 0 } , (2.7)with w = ( w , w ) ∈ X .Note that if w ∈ X is solution of (2.7), then Tr( w ( x, y )) = w ( x, ∈ E is asolution of (1.1), or equivalently, if u ∈ E is a solution of (1.1), then Ext s ( u ) ∈ X is a solution of (2.7).The introduction of this problem is only for the interested reader. As we willsee along the paper, it is not necessary to make use of problem (2.7), i.e., all theresults for (1.1) are going to be proved without using the s -harmonic extensionto the upper half space, E s ( · ). 5or u = ( u, v ) ∈ E , we set I ( u ) = Z R n ( | ( − ∆) s u | + λ u ) dx − Z R n u dx,I ( v ) = Z R n ( | ( − ∆) s v | + λ v ) dx − Z R n v dx, (2.8)Φ( u ) = I ( u ) + I ( v ) − β Z R n u v dx. We also write G β ( u ) = Z R n u dx + Z R n v dx + β Z R n u v dx, and using this notation we can rewrite the energy functional asΦ( u ) = 12 k u k − G β ( u ) , u ∈ E . We observe that G β makes sense because n < s < ⇒ ∗ s > E ֒ → L ( R n ). Even more, any critical point u ∈ E of Φ, gives rise to a solution of (1.1). Definition 2.3
A non-negative critical point e u ∈ E \ { } is called a groundstate of (1.1) if its energy Φ( e u ) is minimal among all the non-trivial criticalpoints of Φ . u ) = ( ∇ Φ( u ) | u ) = ( I ′ ( u ) | u ) + ( I ′ ( v ) | v ) − β Z R n u v dx. We define the Nehari manifold by N = { u ∈ E \ { } : Ψ( u ) = 0 } . Then, one has that( ∇ Ψ( u ) | u ) = −k u k − Z R n u dx < ∀ u ∈ N , (3.1)thus N is a smooth manifold locally near any point u = with Ψ( u ) = 0.Moreover, Φ ′′ ( ) = I ′′ (0) + I ′′ (0) is positive definite, so we infer that is astrict local minimum for Φ. As a consequence, is an isolated point of the set { Ψ( u ) = 0 } , proving that N is a smooth complete manifold of codimension 1,and on the other hand there exists a constant ρ > k u k > ρ ∀ u ∈ N . (3.2)6urthermore, by (3.1) and (3.2) we can show that u ∈ E \ { } is a critical pointof Φ if and only if u ∈ N is a critical point of Φ constrained on N .As a consequence, we have the following. Lemma 3.1 u ∈ E is a non-trivial critical point of Φ if and only if u ∈ N andis a constrained critical point of Φ on N . Remarks 3.2 (i) By the previous arguments, the Nehari manifold N is anatural constraint of Φ . Also, it is relevant to point out that working onthe Nehari manifold, the functional Φ satisfies the following expression, Φ | N ( u ) = 16 k u k + 112 Z R n u dx =: F ( u ) , (3.3) then using (3.2) into (3.3) we obtain Φ( u ) ≥ k u k > ρ ∀ u ∈ N . (3.4) Therefore, by (3.4) the functional Φ is bounded from below on N , as aconsequence we will minimize it on the Nehari manifold. To do so, aremark about compactness is in order.(ii) Analyzing the Palais-Smale (PS) condition, we remember that working onthe radial setting, H = E radial , the embedding of H into L ( R n ) is compactfor n = 2 , , but in dimension n = 1 , the embedding of E or H into L q ( R ) for < q < ∗ s is not compact; see [34, Remarque I.1]. However, wewill analyze all the dimensional cases n = 1 , , , proving that for a PSsequence of Φ on N , we can find a subsequence for which the weak limitis non-trivial and it is a solution of (1.1) . This fact jointly with someproperties of the Schwarz symmetrization will allow us to demonstrate theexistence of positive radially symmetric ground states to (1.1) . Notice thatone could also try to work in the cone of non-negative radially decreasingfunctions, where one has the required compactness, in the one-dimensionalcase, thanks to Berestycki and Lions [10], but this is not our approach. Remark 3.3
It is known; [27, 28], that the equation( − ∆) s v + v = v , (3.5)with v ∈ E , v
0, has a unique radially symmetric and positive solution, thatwe will denote by V . Indeed V is a non-degenerate ground state of (3.5) in H .Clearly, for every β ∈ R , (1.1) already possesses a semi-trivial solution givenby v = (0 , V ) , where V ( x ) = 2 λ V ( λ / s x ) (3.6)is the unique positive radially symmetric solution of ( − ∆) s v + λ v = v in E .7n order to study some useful properties of v , we define define the correspondingNehari manifold associated to I in (2.8), N = { v ∈ E : ( I ′ ( v ) | v ) = 0 } = (cid:26) v ∈ E : k v k − Z R n v dx = 0 (cid:27) . Let us denote T v N the tangent space to N on v . Since h = ( h , h ) ∈ T v N ⇐⇒ ( V | h ) = 34 Z R n V h dx, it follows that ( h , h ) ∈ T v N ⇐⇒ h ∈ T V N . (3.7)Then we prove the following. Proposition 3.4
There exists Λ > such that:(i) if β < Λ , then v is a strict minimum of Φ constrained on N ,(ii) for any β > Λ , then v is a saddle point of Φ constrained on N with inf N Φ < Φ( v ) .Proof . First, we observe that if D Φ N denotes the second derivative of Φ con-strained on N . Using that Φ ′ ( v ) = 0 we have that D Φ N ( v )[ h ] = Φ ′′ ( v )[ h ] for all h ∈ T v N .( i ) We define Λ = inf ϕ ∈ E \{ } k ϕ k R R n V ϕ dx . (3.8)We have that for h ∈ T v N ,Φ ′′ ( v )[ h ] = k h k + I ′′ ( V )[ h ] − β Z R n V h dx. (3.9)Let us take h = ( h , h ) ∈ T v N , by (3.7) h ∈ T V N , then using that V isthe minimum of I on N , there exists a constant c > I ′′ ( V )[ h ] ≥ c k h k . (3.10)Due to (3.10) jointly with (3.9), for β < Λ, there exists another constant c > ′′ ( v )[ h ] ≥ c ( k h k + k h k ) , (3.11)which proves that v is a strict local minimum of Φ on N .( ii ) According to (3.7), h = ( h , ∈ T v N for any h ∈ E . We have that,for β > Λ, there exists e h ∈ E withΛ < k e h k R R n V e h dx < β, h = ( e h, ∈ T v N , by (3.9) we findΦ ′′ ( v )[ h ] = k e h k − β Z R n V e h dx < . (3.12)On the other hand, by (3.7), and using again that V is the minimum of I on N , we have that there exists c > I ′′ ( V )[ h ] ≥ c k h k , ∀ h ∈ T V N . Finally, by (3.9), Φ ′′ ( v )[(0 , h )] = I ′′ ( V )[ h ] for any h ∈ T V N . Thus we havethat there exists a constant c > ′′ ( v )[ h ] ≥ c k h k , ∀ h = (0 , h ) ∈ T v N . The first result on the existence of ground states is given for the coupling pa-rameter β >
Λ in the following.
Theorem 4.1
Assume β > Λ , then Φ has a positive radially symmetric groundstate e u , and there holds Φ( e u ) < Φ( v ) .Proof . By the Ekeland’s variational principle; [26], there exists a PS sequence { u k } k ∈ N ⊂ N , i.e., Φ( u k ) → c N = inf N Φ (4.1) ∇ N Φ( u k ) → . (4.2)By (3.3) and (4.1), we find that { u k } is a bounded sequence on E , hence for asubsequence, we can assume that u k ⇀ u weakly in E , (4.3) u k → u strongly in L qloc ( R ) = L qloc ( R ) × L qloc ( R ) ∀ ≤ q < ∗ s , (4.4)and also u k → u a. e. in R n . Since N is closed we have that u ∈ N , evenmore, using that is an isolated point the set { Ψ( u ) = 0 } we infer that u = .On the other hand, the constrained gradient satisfies ∇ N Φ( u k ) = Φ ′ ( u k ) − η k Ψ ′ ( u k ) → , (4.5)where η k is the corresponding Lagrange multiplier. Taking the scalar productwith u k in (4.5), since u k ∈ N we have that (Φ ′ ( u k ) | u k ) = Ψ( u k ) = 0, thenwe infer that η k (Ψ ′ ( u k ) | u k ) →
0; this jointly with (3.1),(3.3) and the fact that k Ψ ′ ( u k ) k ≤ C < ∞ imply that η k → ′ ( u k ) → u k → u in E , we infer that u ∈ N is a non-trivial critical point of Φ and byLemma 3.1 it is also a non-trivial critical point of Φ on N .Moreover, using that u ∈ N jointly with (3.3) and the Fatou’s Lemma, wefind Φ( u ) = F ( u ) ≤ lim inf k →∞ F ( u k )= lim inf k →∞ Φ( u k ) = c N . As a consequence, u is a least energy solution of (1.1). By Proposition 3.4-( ii ) we know that necessarily Φ( u ) < Φ( v ). Additionally, by the maximumprinciple in the fractional setting; [13], applied to the second equation in (1.1),we have that v >
0. In order to show that also u >
0, first we prove thefollowing.
Claim.
We can assume without loss of generality that u ≥ | u | = ( | u | , v ), then we have two cases:1. If | u | ∈ N , by the Stroock-Varopoulos inequality; [41, 42], k ( − ∆) s ( | u | ) k L ≤ k ( − ∆) s ( u ) k L , (4.6)we have, in particular, that k| u |k ≤ k u k , then we obtainΦ( | u | ) ≤ Φ( u ) = c N . Then, by similar arguments as in [43, Theorem 4.3], we have that | u | isa non-negative ground state.2. If | u | 6∈ N , we take the unique t > t = 1 such that t | u | ∈ N , whichcomes from k | u | k = t Z R n u dx + t (cid:18) Z R n v dx + 32 β Z R n u v dx (cid:19) . (4.7)Since u ∈ N , then k u k = Z R n u dx + 12 Z R n v dx + 32 β Z R n u v dx. (4.8)By (4.7), (4.8) and again the Stroock-Varopoulos inequality (4.6), we inferthat t Z R n u dx + t (cid:18) Z R n v dx + 32 β Z R n u v dx (cid:19) ≤ Z R n u dx + 12 Z R n v dx + 32 β Z R n u v dx. (4.9)10sing that t = 1, as a consequence of (4.9) we deduce that 0 < t < t < t | u | ) = t k | u | k + t Z R n u dx< k | u | k + 112 Z R n u dx ≤ Φ( u ) = c N . This is a contradiction because t | u | ∈ N . Therefore | u | ∈ N and theclaim is proved.Once we can assume without loss of generality that u ≥
0, by the maximumprinciple applied to the first equation in (1.1) we find u > u is a positive ground state.To finish the proof, we have to show that the ground state is indeed radiallysymmetric.If u is not radially symmetric, we set e u = u ⋆ = ( u ⋆ , v ⋆ ), where u ⋆ , v ⋆ de-note the Schwarz symmetric functions associated to u , v respectively. By theproperties of the Schwarz symmetrization; see for instance [30] for the fractionalsetting and [8] for the classical one, there hold k u ⋆ k ≤ k u k , G β ( u ⋆ ) ≥ G β ( u ) . (4.10)Furthermore, there exists a unique t ⋆ > t ⋆ e u ∈ N . If t ⋆ = 1, by(4.10) we have Φ( e u ) ≤ Φ( u ) = c N with e u ∈ N thus e u is a positive radiallysymmetric ground state of (1.1).On the contrary, i.e., if t ⋆ = 1, as in (4.7), t ⋆ comes from k e u k = t ⋆ Z R n ( u ⋆ ) dx + t ⋆ (cid:18) Z R n ( v ⋆ ) dx + 32 β Z R n ( u ⋆ ) v ⋆ dx (cid:19) . (4.11)Due to u ∈ N , (4.10), (4.11), the fact that u > and t ⋆ > Z R n u dx + 12 Z R n v dx + 32 β Z R n u v dx ≥ t ⋆ Z R n u dx + t ⋆ (cid:18) Z R n v dx + 32 β Z R n u v dx (cid:19) . (4.12)Thus, using that 0 < t ⋆ = 1 in (4.12), we obtain 0 < t ⋆ <
1, this and (4.10)show thatΦ( t ⋆ e u ) = 16 t ⋆ k u ⋆ k + 112 t ⋆ Z R n ( u ⋆ ) dx < k u k + 112 Z R n u dx = Φ( u ) = c N , (4.13)with t ⋆ e u ∈ N which is a contradiction with (4.13), proving that t ⋆ = 1 and asabove, then we finish the proof. The second result about existence of groundstates cover the range 0 < β ≤ Λ, provided λ is large enough.11 heorem 4.2 There exists Λ > such that if λ > Λ , System (1.1) has aradially symmetric ground state e u > for every < β ≤ Λ .Proof . Arguing in the same way as in the proof of Theorem 4.1, we prove thatthere exists a radially symmetric ground state e u ≥ . Moreover, in Theorem4.1 for β > Λ we proved that e u > . Now we need to show that for 0 < β ≤ Λindeed e u > which follows by the maximum principle provided e u = v . Takinginto account Proposition 3.4-( i ), v is a strict local minimum of Φ on N , andthis does not guarantee that u v . Following [19], the idea consists on theconstruction of a function u = ( u , v ) ∈ N with Φ( u ) < Φ( v ). To do so,since v = (0 , V ) is a local minimum of Φ on N provided 0 < β < Λ, we cannotfind u in a neighborhood of v on N . Thus, we define u = t ( V , V ) where t > u ∈ N .Now, we will show that u = t ( V , V ) ∈ N with Φ( u ) < Φ( v ) , for λ large enough.Notice that t > u ) = 0, i.e., t k ( V , V ) k − t Z R n V dx − t (1 + 3 β ) Z R n V dx = 0 . (4.14)We also have k ( V , V ) k = 2 k V k + ( λ − λ ) Z R n V dx. (4.15)Moreover, since V ∈ N , we have k V k − Z R n V dx = 0 . (4.16)Substituting (4.15) and (4.16) in (4.14) it follows t (cid:18)Z R n V dx + ( λ − λ ) Z R n V dx (cid:19) − t Z R n V dx − t (1+3 β ) Z R n V dx = 0 . (4.17)Hence, applying the scaling (3.6) yields Z R n V r dx = 2 r λ r − n s Z R n V r dx. (4.18)Subsequently, substituting (4.18) for r = 2 , , λ − n s t we have that Z R n V dx + λ − λ λ Z R n V dx − λ t Z R n V dx − t (1 + 3 β ) Z R n V dx = 0 . (4.19)12oreover, by (3.3), (4.15) and (4.16) we find respectively the expressionsΦ( u ) = t (cid:18)Z R n V dx + ( λ − λ ) Z R n V dx (cid:19) + t Z R n V dx, (4.20)Φ( v ) = I ( V ) = k V k − Z R n V = Z R n V . (4.21)By (4.20), (4.21) we have Φ( u ) < Φ( v ) is equivalent to t (cid:18)Z R n V dx + ( λ − λ ) Z R n V dx (cid:19) + t Z R n V dx − Z R n V dx < , (4.22)and then, applying again (4.18) and multiplying (4.22) by 6 λ n s − , we actuallyhave t (cid:18)Z R n V dx + λ − λ λ Z R n V dx (cid:19) + t λ Z R n V dx − Z R n V dx < . (4.23)For λ large enough we find that (4.19) will provide us with (4.23). Therefore,there exists a positive constant Λ such that for λ > Λ inequality (4.23) holds,proving that Φ( e u ) ≤ Φ( u ) < Φ( v ) . Finally, this shows that e u = v and we finish. equations In this last subsection, we deal with some extended systems of (1.1) to morethan two equations.We start with the study of the following system coming from NLFS-2FKdVequations if n = 1 or 3NLFS equations if n = 1 , , ( − ∆) s u + λ u = u + β uv + β uv , ( − ∆) s v + λ v = v + β u , ( − ∆) s v + λ v = v + β u , (5.1)where u, v , v ∈ E . This system can be seen as a perturbation of (1.1) when | β | or | β | is small.We use similar notation as in previous sections with natural meaning, forexample, E = E × E × E , = (0 , , u ) = 12 k u k − Z R n u dx − Z R n ( v + v ) dx − Z R n u ( β v + β v ) dx (5.2) N = { u ∈ E \ { } : (Φ ′ ( u ) | u ) = 0 } , (5.3)etc.Let U ∗ , V ∗ j be the unique positive radially symmetric solutions of ( − ∆) s u + λ u = u , ( − ∆) s v + λ j v = v in E respectively, j = 1 ,
2; see [27, 28].13 emark 5.1
The unique non-negative semi-trivial solutions of (5.1) are givenby v ∗ = (0 , V ∗ , v ∗ = (0 , , V ∗ ) and v ∗ = (0 , V ∗ , V ∗ ).Following Section 4, the first result about existence of ground states is thefollowing. Theorem 5.2
Assume β j > Λ j for j = 1 , , then (5.1) has a positive radiallysymmetric ground state e u .Proof . We define Λ j = inf ϕ ∈ E \{ } k ϕ k R R n V ∗ j ϕ dx j = 1 , . (5.4)where k · k is the norm in E with λ .As in Proposition 3.4-( ii ), using that β j > Λ j , j=1, 2, one can show thatboth v ∗ , v ∗ are saddle points of the energy functional Φ (defined by (5.2))constrained on the Nehari manifold N (defined by (5.2)). Then c N = inf N Φ < min { Φ( v ∗ ) , Φ( v ∗ ) } < Φ( v ∗ ) = Φ( v ∗ ) + Φ( v ∗ ) . (5.5)By the Ekeland’s variational principle, there exists a PS sequence { u k } k ∈ N ⊂ N ,i.e., Φ( u k ) → c N (5.6) ∇ N Φ( u k ) → . (5.7)The lack of compactness can be circumvent arguing in a similar way as in theproof of Theorem 4.1, proving that for a subsequence, u k ⇀ e u weakly in E with e u (cid:1) , e u ∈ N a critical point of Φ satisfying Φ( e u ) = c N , then e u is anon-negative ground state.To prove the positivity of e u , if one supposes that the first component u ∗ ≡
0, since the only non-negative solutions of (5.1) are the semi-trivial solutionsdefined in Remark 5.1, we obtain a contradiction with (5.5). Furthermore, ifthe second or third component vanish then e u must be , and this is not possiblebecause Φ | N is bounded bellow by a positive constant like in (3.4), then is anisolated point of the set { u ∈ E : Ψ( u ) = (Φ ′ ( u ) | u ) = 0 } , proving that N is acomplete manifold, as in the previous sections. Then, the maximum principleshows that e u > . Finally, to show that we have a radially symmetric groundstate, we argue as in the proof of Theorem 4.1.Furthermore, following the ideas in the proof of Theorem 4.2 we have thefollowing.
Theorem 5.3
Assume that β , β > (but not necessarily β j > Λ j as in Theo-rem 5.2). Then there exists a positive radially symmetric ground state e u provided λ , λ are sufficiently large. roof . The proof follows the same ideas as the one of Theorem 4.2 with ap-propriate changes. For example, in order to prove the positivity, one has toshow that there exists u ∈ N with Φ( u ) < min { Φ( v ∗ ) , Φ( v ∗ ) } , that holdstrue provided λ , λ are large enough. We omit details for short.Plainly we can extend these results to systems with an arbitrary number ofequations N > ( − ∆) s u + λ u = u + N − X k =1 β k uv k ( − ∆) s v j + λ j v j = 12 v j + 12 β j u ; j = 1 , · · · , N − . (5.8)Arguing as in Theorems 5.2, 5.3 we can show the next result. Theorem 5.4
There exists a positive radially symmetric ground state of (5.8) if • either β k > Λ k = inf ϕ ∈ E \{ } k ϕ k R R n V ∗ k ϕ dx ; k = 1 , · · · N − , where V ∗ k denotes the unique positive radial solution of ∆ v + λ k v = v in E ; k = 1 , · · · , N − , • or β j > are arbitrary and λ j are large enough; j = 1 , . . . , N − . Remark 5.5
As was commented in [19] for the local setting, here in the non-local fractional framework, another natural extension of (1.3) to more than twoequations different from (5.1) is the following system coming from 2NLFS-FKdVequations if n = 1 or 3NLFS equations if n = 1 , , ( − ∆) s u + λ u = u + β u u + β u v ( − ∆) s u + λ u = u + β u u + β u v ( − ∆) s v + λv = v + β u + β u . (5.9)We denote U j the unique positive radially symmetric solution of ( − ∆) s u + λ j u = u in E ; j = 1 ,
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