A minimization method and applications to the study of solitons
aa r X i v : . [ m a t h . A P ] N ov A minimization method and applications to thestudy of solitons
Vieri Benci ∗ Donato Fortunato † Dedicated to V. Lakshmikantham
November 3, 2018
Abstract
Roughly speaking a solitary wave is a solution of a field equation whoseenergy travels as a localized packet and which preserves this localizationin time. A soliton is a solitary wave which exhibits some strong form ofstability so that it has a particle-like behavior. In this paper, we provea general, abstract theorem (Theorem 26) which allows to prove the ex-istence of a class of solitons. Such solitons are suitable minimizers of aconstrained functional and they are called hylomorphic solitons. Then weapply the abstract theory to problems related to the nonlinear Schr¨odingerequation (NSE) and to the nonlinear Klein-Gordon equation (NKG).
AMS subject classification:
Key words: lack of compactness, orbital stability, nonlinear Schr¨odingerequation, lattice, nonlinear Klein-Gordon equation, solitary waves, hylo-morphic solitons, vortices.
Contents ∗ Dipartimento di Matematica Applicata, Universit`a degli Studi di Pisa, Via F. Buonarroti1/c, Pisa, ITALY and Department of Mathematics, College of Science, King Saud University,Riyadh, 11451, SAUDI ARABIA. e-mail: [email protected] † Dipartimento di Matematica, Universit`a degli Studi di Bari Aldo Moro, Via Orabona 4,70125 Bari, Italy, e-mail: [email protected] An existence result of vortices for NSE 9
In some recent papers ([4], [6], [9], [14], [11], [12]) the existence of solitonshas been proved using variational methods. In this paper, we prove a general,abstract theorem (Theorem 26) which applies to most of the situations analyzedin the mentioned papers.The proof of Theorem 26 is carried out in two steps: in the first step (section2) the research of the minimizers of a constrained functional is reduced to thestudy of the minimizers of a suitable free functional. The existence of suchminimizers is stated in Theorem 8. In the second step stability properties ofthese minimizers are proved (section 4).These two theorems (Theorem 8 and Theorem 26) can be applied to thesituations described in the quoted papers relative to solitons. In section 4,we give an abstract definition of soliton (Definition 22) and hylomorphic soliton (Definition 25). These solitons are stable minimizers of a constrained functional.Then we apply the abstract theorems 26 and 8 to new problems related tothe nonlinear Schr¨odinger equation (NSE) and to the nonlinear Klein-Gordonequation (NKG): in section 3 we use Theorem 8 in order to prove the existenceof vortices for the NSE with a potential V which is periodic in one direction(Theorem 12). In section 5 we use Theorem 26 in order to prove the existenceof hylomorphic solitons for NSE in a lattice, namely in presence of a periodicpotential V .Finally in section 6 we use Theorem 26 to prove the existence of hylomorphicsolitons for the nonlinear Klein-Gordon equation (NKG). Let E and C be two functionals on an Hilbert space X. We are interested inthe following minimization problem: find values of σ ∈ R , such that E attains2 constrained minimum on M σ where M σ := { u ∈ X | | C ( u ) | = σ } . In the next section, we will describe the abstract framework where we will workand then we will prove an abstract existence theorem.
We need some definitions. These definitions are related to a couple (
X, G )where G is a group acting on the Hilbert space X. In our applications G will bea subgroup of the group of translations. Definition 1
A non-empty subset Γ ⊂ X is called G -invariant if ∀ u ∈ Γ , ∀ g ∈ G, g u ∈ Γ . Definition 2
A functional J on X is called G -invariant if ∀ g ∈ G, ∀ u ∈ X, J ( g u ) = J ( u ) . Definition 3
A closed G -invariant set Γ ⊂ X is called G -compact if for anysequence u n in Γ there is a sequence g n ∈ G, such that g n u n has a convergingsubsequence. Clearly a sequence u n in X will be called G -compact if its imageis G -compact. Definition 4 A G -invariant functional J on X is called G − compact if anyminimizing sequence u n is G − compact. Clearly a G -compact functional has a G -compact set of minimizers.In order to prove an existence result for the minimizers of E on M σ , weimpose some assumptions to E and C ; to do this we need some other definitions: Definition 5 ( Splitting property)
We say that a functional F on X has thesplitting property if given a sequence u n = u + w n in X such that w n convergesweakly to , we have that F ( u n ) = F ( u ) + F ( w n ) + o (1) . (1) Remark 6
A symmetric, continuous quadratic form satisfies the splitting prop-erty; in fact, in this case, we have that F ( u ) := h L u , u i for some continuousselfajoint operator L ; then given a sequence u n = u + w n with w n ⇀ weakly,we have that F ( u n ) = h L u , u i + h L w n , w n i + 2 h L u , w n i = F ( u ) + F ( w n ) + o (1) . Now we can formulate the required properties on E and C :3 (EC-1) (Value at 0) E, C are C , bounded functionals such that E (0) = 0 , C (0) = 0; E ′ (0) = 0; C ′ (0) = 0 . • (EC-2) (Invariance) E and C are G -invariant. • (EC-3) (Coercivity) We distinguish two cases: C ≥ and C not posi-tive. If C ≥ we assume that there exists a ≥ and s ≥ such that – (i) E ( u ) + aC ( u ) s ≥ – (ii) if k u k → ∞ , then E ( u ) + aC ( u ) s → ∞ ; – (iii) for any bounded sequence u n in X such that E ( u n )+ aC ( u n ) s → , we have that u n → . In the case in which C is not positive we assume that (i), (ii), (iii) holdtrue with a = 0 . • (EC-4) (Splitting property) E and C satisfy the splitting property. Before stating our main theorem we need this definition:
Definition 7
A bounded sequence u n in X is called vanishing sequence iffor any subsequence u n k and any sequence g k ⊂ G , g k u n k converges weakly to . Observe that the notion of vanishing sequence depends on the group G actingon X. Clearly a bounded sequence u n in X is a non-vanishing sequence if thereexists a subsequence u n k and a sequence g k ⊂ G , such that g k u n k convergesweakly to some u = 0 . So, if u n → u n is a vanishing sequence. However, if u n ⇀ We now set Λ( u ) := E ( u ) | C ( u ) | and Λ := inf { lim inf Λ( u n ) | u n is a vanishing sequence } . (2)Now we state and prove the following existence result: Theorem 8
Let E and C be two functionals on a Hilbert space X and G be agroup acting on X. Assume that E and C satisfy (EC-1),...,(EC-4) and infΛ( u ) < Λ . (3)4 hen there is ¯ δ > and a family of values c δ , δ ∈ (cid:0) , ¯ δ (cid:1) , such that the minimum e δ = min { E ( u ) | | C ( u ) | = c δ } (4) exists and the set Γ c δ of minimizers is G -compact. Moreover Γ c δ can be char-acterized as the set of minimizers of the functional J δ ( u ) = Λ ( u ) + δ Φ( u ) δ ∈ (cid:0) , ¯ δ (cid:1) where Φ( u ) = E ( u ) + 2 a | C ( u ) | s with a as in (EC-3). Remark 9
When we will apply Th. 8, it is necessary to estimate Λ ; the fol-lowing inequalities may help to do this. In order to give an estimate from below,assume that there exists a seminorm k u k ♯ such that { u n is a vanishing sequence } ⇒ k u n k ♯ → . (5) Then we have that Λ ≥ lim inf k u k ♯ → Λ( u ) . (6) Let kk denote the norm in X. Since k u n k → ⇒ { u n is a vanishing sequence } , we have that Λ ≤ lim inf k u k→ Λ( u ) . (7) Moreover, if E and C are twice differentiable in , by (EC-0), we have that lim inf k u k→ Λ( u ) = lim inf k u k→ E (0) + E ′ (0) [ u ] + E ′′ (0) [ u , u ] + o ( k u k ) (cid:12)(cid:12)(cid:12) C (0) + C ′ (0) [ u ] + C ′′ (0) [ u , u ] + o ( k u k ) (cid:12)(cid:12)(cid:12) = inf E ′′ (0) [ u , u ] | C ′′ (0) [ u , u ] | and hence Λ ≤ inf E ′′ (0) [ u , u ] | C ′′ (0) [ u , u ] | . It is can be seen that in many applications the two limits (6) and (7) coincideand in this case we get a sharp estimate for Λ . Remark 10 If X = H ( R N ) and G = Z N with the action (94), then (seeProposition 32) the norm k u k ♯ = k u k L t , t ∈ (cid:16) , NN − (cid:17) , N ≥ , satisfies theproperty (5) of the preceding remark 9 and consequently it satisfies also (6)). Remark 11
The fact that the minimization problem of E on M σ reduces to theminimization of a free functional J δ ( u ) is very useful in numerical simulation. roof of theorem 8. By (3) there exists v ∈ X such that Λ( v ) < Λ , thenwe can take δ > J δ ( v ) = Λ ( v ) + δ Φ( v ) < Λ and define ¯ δ = sup { δ | ∃ v : Λ ( v ) + δ Φ( v ) < Λ } . Then inf u ∈ X J δ ( u ) < Λ for δ ∈ (cid:0) , ¯ δ (cid:1) . (8)Now we show that J δ ( u ) ≥ δ u ) − M (9)where M is a suitable constant. Clearly, if C is not positive, we have a = 0in (EC-3)(i); then Φ( u ) = E ( u ) ≥ J δ ( u ) = E ( u ) | C ( u ) | + δE ( u ) . Then (9) isobviously satisfied.Now assume that C ( u ) ≥ . By (EC-3)(i) we have that E ( u ) ≥ − aC ( u ) s (10)and hence E ( u ) C ( u ) ≥ − aC ( u ) s − . (11)Then, by (10) and (11), we get J δ ( u ) = E ( u ) C ( u ) + δ Φ( u ) ≥ − aC ( u ) s − + δ E ( u ) + 2 aC ( u ) s ] + δ u ) ≥ − aC ( u ) s − + δ − aC ( u ) s + 2 aC ( u ) s ] + δ u ) ≥ − aC ( u ) s − + aδ C ( u ) s + δ u ) ≥ δ u ) − M where M = − a min t ≥ (cid:18) δ t s − t s − (cid:19) . Then (9) has been proved.Now let us prove that J δ is G − compact (see Definition 4).Let u n be a minimizing sequence of J δ . This sequence u n is bounded in X. In fact, arguing by contradiction, assume that, up to a subsequence, k u n k → ∞ . Then by (9) and (EC-3) (ii) we get J δ ( u n ) → ∞ which contradicts the fact that u n is a minimizing sequence of J δ . Since u n is minimizing for J δ , by (8), there exists η > n sufficiently large, E ( u n ) | C ( u n ) | + δ Φ( u n ) < Λ − η u n ) = E ( u n ) | C ( u n ) | < Λ − η. (12)On the other hand Λ( u n ) is bounded below. (13)In fact: since Φ is bounded and u n is bounded in X , Φ( u n ) is bounded. So wededuce from (9) that J δ ( u n ) is bounded below. Then Λ( u n ) = J δ ( u n ) − δ Φ( u n )is bounded below and so (13) is proved. By (12) and (13) we have, for somesubsequence, that Λ( u n ) → λ, − ∞ < λ < Λ . (14)Then, by (2), u n is a bounded non vanishing sequence. Hence, by Def. 7, wecan extract a subsequence u n k and we can take a sequence g k ⊂ G such that u ′ k := g k u n k is weakly convergent to some ¯u = 0 . (15)We can write u ′ n = ¯u + w n with w n ⇀ w n → ¯u + w n ) ≥ Φ( ¯u ) + lim Φ( w n ) . (16)If C ( u ) is not positive we have a = 0 in (EC-3)(i), then Φ( u ) = E ( u ) andclearly (16) holds as an equality since E satisfies assumption (EC-4) (splittingproperty).Now assume that C ( u ) ≥ . Then by (EC-4) and since s ≥ , we have thatlim Φ( ¯u + w n ) = lim ( E ( ¯u + w n ) + 2 aC ( ¯u + w n ) s )= E ( ¯u ) + lim E ( w n ) + 2 a lim ( C ( ¯u ) + C ( w n )) s ≥ E ( ¯u ) + lim E ( w n ) + 2 a lim ( C ( ¯u ) s + C ( w n ) s )= E ( ¯u ) + 2 aC ( ¯u ) s + lim E ( w n ) + 2 a lim C ( w n ) s = Φ( ¯u ) + lim Φ( w n ) . (17)So (16) has been proved.Next we show that C ( ¯u + w n ) does not converge to 0 . (18)Arguing by contradiction assume that C ( ¯u + w n ) converges to 0 . Then, since ¯u + w n is a minimizing sequence for J δ , also E ( ¯u + w n ) converges to 0 and then E ( ¯u + w n )+ a | C ( ¯u + w n ) | s → . So, by (EC-3)(iii), we get ¯u + w n → X. (19)7rom (19) and since w n ⇀ X, we have that ¯u = 0, contradicting(15). Then (18) holds and consequently, up to a subsequence, we have | C ( ¯u + w n ) | = | C ( ¯u ) + C ( w n ) + o (1) | ≥ const. > . (20)Now, we set j δ = inf J δ = lim J δ ( u ′ n ); e δ = E ( ¯u ); c δ = | C ( ¯u ) | e = lim E ( w n ); c = lim | C ( w n ) | . Observe that the limits lim E ( w n ) and lim | C ( w n ) | exist (up to subsequences),since E and C are bounded functionals and w n weakly converges.Now we have lim E ( ¯u ) + E ( w n ) + o (1) | C ( ¯u ) + C ( w n ) + o (1) | ≥ e δ + e c δ + c . (21)In fact, as usual we distinguish two cases: if C ≥ C ( ¯u ) + C ( w n ) + o (1)) = c δ + c . On the other hand, if C is not positive, by (EC-3)(i), we have E ≥ | C ( ¯u ) + C ( w n ) + o (1) | ≤ c δ + c . So (21) holds also when C is not positive.Now by (21) and (16), we have that j δ = lim (cid:20) E ( u ′ n ) | C ( u ′ n ) | + δ Φ( u ′ n ) (cid:21) = lim E ( ¯u ) + E ( w n ) + o (1) | C ( ¯u ) + C ( w n ) + o (1) | + δ lim Φ( ¯u + w n ) ≥ e δ + e c δ + c + δ lim Φ( w n ) + δ Φ( ¯u ) . (22)Now we want to prove that e c ≥ e δ c δ . (23)We argue indirectly and we suppose that e δ c δ > e c . (24)By the above inequality it follows that e δ + e c δ + c = e δ c δ c δ + e c c c δ + c > e c c δ + e c c c δ + c = e c (25)and hence, by (22) and (25), we get j δ ≥ e δ + e c δ + c + δ lim Φ( w n ) + δ Φ( ¯u )8 e c + δ lim Φ( w n ) + δ Φ( ¯u ) = lim J δ ( w n ) + δ Φ( ¯u ) ≥ inf J δ + δ Φ( ¯u ) = j δ + δ Φ( ¯u ) ≥ j δ . So we get a contradiction and (24) cannot occur. Then we have (23).Now, by (23), we get e δ + e c δ + c = e δ c δ c δ + e c c c δ + c ≥ e δ c δ c δ + e δ c δ c c δ + c = e δ c δ . So we get e δ + e c δ + c ≥ e δ c δ . Then, using (22), the above inequality and the fact that j δ = inf J δ , we get j δ ≥ e δ + e c δ + c + δ Φ( ¯u ) + δ lim Φ( w n ) ≥ e δ c δ + δ Φ( ¯u ) + δ lim Φ( w n )= J δ ( ¯u ) + δ lim Φ( w n ) ≥ j δ + δ lim Φ( w n ) . Then δ lim Φ( w n ) ≤ w n → u ′ n → ¯u strongly and ¯u is a minimizerof J δ . So we conclude that J δ is G − compact.Since ¯u is a minimizer of J δ , clearly ¯u minimizes also the functional E ( u ) c δ + δ [ E ( u ) + ac sδ ] = (cid:18) c δ + δ (cid:19) E ( u ) + δac sδ on the set { u ∈ X | | C ( u ) | = c δ } and hence ¯u minimizes also E on this set.Now denote by Γ c δ the set of such minimizers. It is easy to see that viceversaΓ c δ is contained in the set of minimizers of J δ . So, since J δ is G -compact, weconclude that Γ c δ is G -compact. (cid:3) The existence of vortices is an interesting and old issue in many questions ofmathematical physics as superconductivity, classical and quantum field theory,string and elementary particle theory (see the pioneering papers [1], [25] ande.g. the more recent ones [24], [29], [30], [28] with their references).From mathematical viewpoint, the existence of vortices for the nonlinearKlein-Gordon equations (NKG), for nonlinear Schroedinger equations (NSE)9nd for nonlinear Klein-Gordon-Maxwell equations (NKGM) has been studiedin some recent papers ( [18], [2], [3], [10], [16], [7], [8], [15]).Many of the previous results can be obtained applying Th.8. Here we willconsider a case not covered by the existing literature, namely the study of vor-tices in NSE when the potential V ( x ) depends only on the third variable and itis periodic, namely for all k ∈ Z V ( x , x , x ) = V ( x ) = V ( x + k ) . (26) Let us consider the nonlinear Schroedinger equation: i ∂ψ∂t = −
12 ∆ ψ + 12 W ′ ( ψ ) + V ( x ) ψ. (NSE)where ψ ( t, x ) is a complex valued function defined on the space-time R × R N ( N ≥ , V : R N → R , W : C → R such that W ( ψ ) = F ( | ψ | ) for some smoothfunction F : [0 , ∞ ) → R and W ′ ( ψ ) = ∂W∂ψ + i ∂W∂ψ , ψ = ψ + iψ (27)namely W ′ ( ψ ) = F ′ ( | ψ | ) ψ | ψ | . Equation (NSE) is the Euler-Lagrange equation relative to the Lagrangiandensity L = Re (cid:0) i∂ t ψψ (cid:1) − |∇ ψ | − V ( x ) | ψ | − W ( ψ ) . By the well known Noether’s theorem (see e.g. [22], [9]) the invariance of L under a one parameter Lie group gives rise to a constant of the motion.Since L is invariant under the action of the time translations the energy E ( ψ ) = 12 Z h |∇ ψ | + V ( x ) | ψ | i dx + Z W ( ψ )is constant along the solutions of (NSE).Since W ( ψ ) = F ( | ψ | ) , L is invariant under the S action ψ → e iθ ψ, then the charge C ( ψ ) , defined by C ( ψ ) = Z | ψ | , is constant along the solutions of (NSE) (see e.g. [9]).10he angular momentum, by definition, is the quantity which is preserved byvirtue of the invariance under space rotations (with respect to the origin) of theLagrangian. We shall consider, for simplicity, the case of three space dimensions N = 3 . If we assume that V ( x , x , x ) = V ( x )namely that V depends only on the third coordinate, then the Lagrangian isinvariant under the group of rotations around the axis x . In this case the thirdcomponent of the momentum M ( ψ ) = Re Z ( x ∂ x ψ − x ∂ x ψ ) dx (28)is a constant of motion. Using the polar form ψ ( t, x ) = u ( t, x ) e iS ( t,x ) , u ≥ M ( ψ ) can be written as follows M ( ψ ) = Z ( x ∂ x S − x ∂ x S ) u dx. A solution of (NSE) is called standing wave if it has the following form: ψ ( t, x ) = ψ ( x ) e − iωt , ω > vortex is a standing wave with nonvanishing angular momentum .It is immediate to check that if ψ ( x ) in (30) has real values, the angularmomentum M ( ψ ) is trivial . However, if ψ ( x ) is allowed to have complex val-ues, it is possible to have M ( ψ ) = 0 . Thus, we are led to make an ansatz of thefollowing form: ψ ( t, x ) = u ( x ) e i ( ℓθ ( x ) − ωt ) , u ( x ) ≥ , ω ∈ R , ℓ ∈ Z − { } (31)and θ ( x ) = Im log( x + ix ) ∈ R / π Z ; x = ( x , x , x ) . Moreover, we assume that u has a cylindrical symmetry, namely u ( x ) = u ( r, x ) , where r = q x + x . (32)By this ansatz, equation (NSE) is equivalent to the system (cid:26) −△ u + ℓ |∇ θ | u + W ′ ( u ) + 2 V ( x ) ψ = 2 ωuu △ θ + 2 ∇ u · ∇ θ = 0 . By the definition of θ and (32) we have △ θ = 0 , ∇ θ · ∇ u = 0 , |∇ θ | = 1 r · denotes the euclidean scalar product.So the above system reduces to find solutions, with symmetry (32), of theequation − △ u + ℓ r u + W ′ ( u ) + 2 V ( x ) u = 2 ωu in R . (33)Direct computations show that the energy and the third component of the an-gular momentum become E ℓ ( u ) = E (cid:16) u ( x ) e i ( ℓθ ( x ) − ωt ) (cid:17) (34)= Z R (cid:20) |∇ u | + (cid:18) ℓ r + V ( x ) (cid:19) u + W ( u ) (cid:21) dx (35) M (cid:16) u ( x ) e i ( ℓθ ( x ) − ωt ) (cid:17) = − ℓ Z R u dx. (36)We point out that M in (36) is nontrivial when both ℓ and u are not zero.Let us remark that the solutions of equation (33) can be obtained as criticalpoints of the functional (34) on the manifold M c := { u ∈ X | C ( u ) = c } where X is the Hilbert space obtained by the closure of D ( R N ) with respectto the norm k u k X = Z R (cid:20) |∇ u | + (cid:18) ℓr + 1 (cid:19) u (cid:21) dx. (37)Thus we can apply the minimization result (Theorem 8) stated in section 2.Cleary, using this approach, 2 ω will be the Lagrange multiplier. Recall that W ( ψ ) = F ( | ψ | ) . With abuse of notation, in the following we write W instead of F We make the following assumptions:(i) W : R + → R is a C function which satisfies the following assumptions: W (0) = W ′ (0) = W ′′ (0) = 0 ( W ) | W ′ ( s ) | ≤ c s r − + c s q − for q, r in (2 , ∗ ) , ∗ = 2 NN − , N ≥ W ) W ( s ) ≥ − cs p , c ≥ , < p < N for s large ( W ) ∃ s ∈ R + such that W ( s ) s < inf V − sup V ( W ) D ( R N ) denotes the space of the infinitely differentiable functions with compact supportdefined in R N . V : R N → R is a continuous function which satisfies the following assump-tions: 1 ≤ V ( x ) ≤ V < ∞ ; ( V ) ∀ k ∈ Z , V ( x ) = V ( x ) = V ( x + k ) . ( V )We get the following theorem: Theorem 12
Assume that (W ) , ...,(W ) and ( V ), ( V ) , are satisfied. Then,for any integer ℓ = 0 , there exist ¯ δ > and a family ψ δ , δ ∈ (0 , ¯ δ ) , of vorticesof (NSE) with angular momentum (cid:16) , , − ℓ R R | ψ δ | dx (cid:17) . Remark 13
The conditions ( W ) and ( V ) are assumed for simplicity; in factthey can be easily weakened as follows W (0) = W ′ (0) = 0 , W ′′ (0) = E and E ≤ V ( x ) ≤ V < + ∞ . In fact, in the general case, it is possible to replace W ( s ) with W ( s ) = W ( s ) − E s and V ( x ) with V ( x ) = V ( x ) − E + 1 . In this case equation (33) becomes − △ u + ℓ r u + W ′ ( u ) + 2 V ( x ) u = ( − E − E + 2 + 2 ω ) u in R . (38) Thus in the general case, there is only a change of the lagrange multiplier andso the solution of the Schroedinger equation is modified only by a phase factor.
By the preceding subsection we deduce that the existence of vortices ofangular momentum ℓ is reduced to find critical points, having the symmetry(32), of the functional E ℓ (34) on M c .Now consider the action T θ of the group S on u ( x , x , x ) ∈ X, defined by T θ u = u ( R θ ( x , x ) , x ) , θ ∈ R π Z , (39)where R θ denotes the rotation of an angle θ in the plane x , x . We set X r = { u ∈ X | u = u ( r, x ) } , M rc = M c ∩ X r . Observe that V depends only on x , then the functional E ℓ is invariant underthe action (39). So by the Palais principle of symmetric criticality [26], thecritical points of E ℓ on M rc are also critical points of E ℓ on M c ;moreover thesecritical points clearly have the symmetry (32).These observations show that the proof of Theorem 12 is an immediateconseguence of the following proposition:13 roposition 14 Let the assumptions of Theorem 12 be satisfied. Then , forany integer ℓ, there exist ¯ δ > and a family of values of charges c δ , δ ∈ (0 , ¯ δ ) , such that E ℓ possesses a minimizer on any M rc δ . In order to prove Proposition 14 we shall use Theorem 8. In this case wehave u = u, u ∈ X r and E ( u ) = E ℓ ( u ) , C ( u ) = C ( u ) = Z u dx where E ℓ ( u ) is defined in (34).Observe first that, by ( V ), E ℓ is invariant under the action T k of the group G = Z on X r defined by T k u ( r, x ) = u ( r, x + k ) , k ∈ Z Clearly E ℓ and C satisfy assumptions (EC-1), (EC-2).In the following Lemmas we shall show that E ℓ and C satisfy also (EC-3),(EC-4) and (3). Lemma 15
Let the assumptions of Theorem 12 be satisfied. Then E ℓ and C satisfy the coercivity assumption (EC-3). Proof.
We recall a well known inequality: there exists a constant b p > , such that for any u in H ( R N ) ( N ≥ || u || L p ≤ b p || u || − N ( − p ) L ||∇ u || N ( − p ) L . (40)Then || u || pL p ≤ b p || u || p − pN ( − p ) L ||∇ u || pN ( − p ) L . (41)Since 2 < p < N , then pN (cid:16) − p (cid:17) := q < . So || u || pL p ≤ b p || u || rL ||∇ u || qL (42)where r = p − pN (cid:16) − p (cid:17) = p − q > . Then by H¨older inequality we have || u || pL p ≤ b p M || u || rL M ||∇ u || qL ≤ γ ′ ( b p M || u || rL ) γ ′ + 1 γ (cid:18) M ||∇ u || qL (cid:19) γ = ( b p M ) γ ′ γ ′ || u || rγ ′ L + 1 γM γ ||∇ u || qγL . Now chose γ = q and M = (cid:16) cγ (cid:17) /γ (where c is the constant in assumption(W )) so that || u || pL p ≤ ( b p M ) γ ′ γ ′ || u || rγ ′ L + 12 c ||∇ u || L . c || u || pL p ≤ a || u || sL + 12 ||∇ u || L (43)where a = c ( b p M ) γ ′ γ ′ ; s = rγ ′ . So we have, taking N = 3 , and using ( W ) and (43) E ℓ ( u ) + aC ( u ) s = 12 ||∇ u || L + Z V u + ℓ Z u r + Z W ( u ) + a || u || sL ≥ ||∇ u || L + Z V u + ℓ Z u r − c Z | u | p + a || u || sL ≥ ℓ Z u r + Z V u . (44)Observe that, since p > , we have s > . So (EC-3)(i) is satisfied. Moreover itcan be easily verifed that also (EC-3)(ii) is satisfied.Now let us prove (EC-3)(iii). Let u n be a bounded sequence in X r such thatΦ( u n ) → , then by (44) we have ℓ Z u n r + Z V u n → . (45)So R u n → ℓ R u n r → . Then, in order to show that u n goes to 0 in X r , it remains to prove that ||∇ u n || L → . (46)Since u n is bounded in X r , by (42) we get Z | u n | p → . (47)Since Φ( u n ) → ), we have0 = lim( E ℓ ( u n ) + aC ( u n ) s ) ≥ lim sup( 12 ||∇ u n || L + D n ) (48)where D n = ℓ Z u n r + Z V u n + a Z u n − c Z | u n | p . (49)By (45) and (47) we get D n → . So by (48) we deduce (46). (cid:3)
Lemma 16
Let the assumptions of theorem 12 be satisfied. Then E ℓ and C satisfy the splitting property (EC-4). roof. Consider any sequence u n = u + w n ∈ X r where w n converges weakly to 0 . We set E ℓ ( v ) = A ( v, v ) + K ( v )where A ( v, v ) = Z R (cid:20) |∇ v | + (cid:18) ℓ r + V ( x ) (cid:19) v (cid:21) and K ( v ) = Z W ( v ) dx. Since C ( v ) = R v and A ( v, v ) are quadratic, by remark 6, we have only to showthat K ( v ) satisfies the splitting property. For any measurable A ⊂ R and any ν ∈ X r , we set K A ( v ) = Z A W ( v ) dx. Choose ε > R = R ( ε ) > (cid:12)(cid:12) K B cR ( u ) (cid:12)(cid:12) < ε (50)where B cR = R N − B R and B R = (cid:8) x ∈ R N : | x | < R (cid:9) . Since w n ⇀ H (cid:0) R (cid:1) , by usual compactness arguments, we havethat K B R ( w n ) → K B R ( u + w n ) → K B R ( u ) . (51)Then, by (50) and (51), we havelim n →∞ | K ( u + w n ) − K ( u ) − K ( w n ) | = lim n →∞ (cid:12)(cid:12) K B cR ( u + w n ) + K B R ( u + w n ) − K B cR ( u ) − K B R ( u ) − K B cR ( w n ) − K B R ( w n ) (cid:12)(cid:12) = lim n →∞ (cid:12)(cid:12) K B cR ( u + w n ) − K B cR ( u ) − K B cR ( w n ) (cid:12)(cid:12) ≤ lim n →∞ (cid:12)(cid:12) K B cR ( u + w n ) − K B cR ( w n ) (cid:12)(cid:12) + ε. (52)Now, by the intermediate value theorem, there exists ζ n ∈ (0 ,
1) such thatfor z n = ζ n u + (1 − ζ n ) w n , we have that (cid:12)(cid:12) K B cR ( u + w n ) − K B cR ( w n ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)D K ′ B cR ( z n ) , u E(cid:12)(cid:12)(cid:12) Z B cR | W ′ ( z n ) u | ≤ (by ( W )) (53) ≤ Z B cR c | z n | r − | u | + c | z n | q − | u | (54) ≤ c k z n k r − L r ( B cR ) k u k L r ( B cR ) + c k z n k q − L q ( B cR ) k u k Lq ( BcR ) (55)(if R is large enough) (56) ≤ c (cid:16) k z n k r − L r ( B cR ) + k z n k q − L q ( B cR ) (cid:17) ε (57)So we have (cid:12)(cid:12) K B cR ( u + w n ) − K B cR ( w n ) (cid:12)(cid:12) ≤ c (cid:16) k z n k r − L r ( B cR ) + k z n k q − L q ( B cR ) (cid:17) ε (58)Since z n is bounded in H (cid:0) R (cid:1) , the sequences k z n k r − L r ( B cR ) and k z n k q − L q ( B cR ) arebounded. Then, by (52) and (58), we easily getlim n →∞ | K ( u + w n ) − K ( u ) − K ( w n ) | ≤ ε + M · ε. (59)where M is a suitable constant.Since ε is arbitrary, from (59) we getlim n →∞ | K ( u + w n ) − K ( u ) − K ( w n ) | = 0 (cid:3) Now in order to prove that assumption (3) is satisfied some work is necessary.Set Λ( u ) = E ℓ ( u ) C ( u ) . First of all we have:
Lemma 17
If the assumptions of Theorem 12 are satisfied, then for > t > , we have lim inf u ∈ X r , k u k Lt → Λ( u ) = inf u ∈ X r k u k Lt =1 12 R (cid:16) |∇ u | + ℓ u r (cid:17) dx + R V u R u . Proof.
Clearly lim inf u ∈ X r , k u k Lt → Λ( u ) = lim inf u ∈ X r , k u k Lt =1 ,ε → E ( εu ) C ( εu )= inf u ∈ X r , k u k Lt =1 , R (cid:16) |∇ u | + ℓ u r (cid:17) dx + R V u R u + lim inf u ∈ X r , k u k Lt =1 ,ε → R W ( εu ) ε R u .
17o the proof of Lemma will be achieved if we show thatlim inf u ∈ X r , k u k Lt =1 ,ε → R W ( εu ) ε R u = 0 . (60)By assumptions ( W ) and ( W ) we have − cs p ≤ W ( s ) ≤ ¯ c ( s q + s r ) (61)where c, ¯ c are positive constants and q, r in (2 , . Then by (61) we have − cAε p − ≤ inf k u k Lt =1 R W ( εu ) ε R u ≤ ¯ cB ( ε q − + ε r − ) (62)where A = inf u ∈ X r k u k Lt =1 R | u | p R u , B = inf u ∈ X r k u k Lt =1 R ( | u | q + | u | r ) R u . By (62) we easily get (60). (cid:3)
Now consider the following action T k of the group G = Z on X r :for all u ∈ X r , k ∈ Z T k u ( x , x , x ) = u ( x , x , x + k )The following proposition holds Lemma 18 If < t < , the norm k u k L t satisfies the property (5), namely { u n is a vanishing sequence } ⇒ k u n k L t → . Proof.
Let u n be a G-vanishing sequence in X r and, arguing by contradic-tion, assume that k u n k L t does not converge to 0 . Then, up to a subsequence, k u n k L t ≥ a > . (63)Since u n is bounded in X r , we have that for a suitable constant M > k u n k H ≤ M. (64)Now we set Q i = { ( x , x , x ) : i ≤ x < i + 1 } , i integer.Clearly R = [ i ∈ Z Q i C denote the constant for the Sobolev embedding H ( Q i ) ⊂ L t ( Q i ) , then,by (63) and (64), we get the following0 < a t ≤ Z | u n | t = X i Z Q i | u n | t = X i k u n k t − L t ( Q i ) k u n k L t ( Q i ) ≤ (cid:18) sup i k u n k t − L t ( Q i ) (cid:19) · X i k u n k L t ( Q i ) ≤ C (cid:18) sup i k u n k t − L t ( Q i ) (cid:19) · X i k u n k H ( Q i ) = C (cid:18) sup i k u n k t − L t ( Q i ) (cid:19) k u n k H ≤ CM (cid:18) sup i k u n k t − L t ( Q i ) (cid:19) . Then (cid:18) sup i k u n k L t ( Q i ) (cid:19) ≥ (cid:18) a t CM (cid:19) / ( t − . So, for any n, there exists an integer i n such that k u n k L t ( Q in ) ≥ α > . (65)Then k T i n u n k L t ( Q ) = k u n k L t ( Q in ) ≥ α > . Since u n and then T in u n is bounded in X r , we have, passing eventually to asubsequence, that T in u n ⇀ u weakly in X r . Clearly, if we show that u = 0 , we get a contradiction with the assumptionthat u n is nonvanishing.Now, let ϕ = ϕ ( x ) be a nonnegative, C ∞ -function whose value is 1 for0 < x < | x | > . Then the sequence ϕT i n u n is bounded in H ( R × ( − , , moreover ϕT i n u n is invariant under the action (39). Then,using the compactness result proved in [21], we have ϕT i n u n → χ strongly in L t ( R × ( − , . On the other hand ϕT i n u n → ϕu a.e . (66)Then ϕT i n u n → ϕu strongly in L t ( R × ( − , . (67)Moreover k ϕT i n u n k L ( R × ( − , ≥ k ϕT i n u n k L t ( Q ) = k u n k L t ( Q in ) ≥ α > . (68)Then by (67) and (68) k ϕu k L ( R × ( − , ≥ α > . u = 0 . (cid:3) Finally it remains to show that also assumption (3) is satisfied.
Lemma 19
Let the assumptions of Theorem 12 be satisfied, then inf u ∈ X r Λ( u ) < Λ . Proof.
We shall show that there exists u ∈ X r such that Λ( u ) < Λ . Theconstruction of such u needs some work since we require that u belongs to X r , namely we require that u is invariant under the S action (39) and it is 0 nearthe x axis, so that R u r converges.For 0 < µ < λ we set: T λ,µ = (cid:8) ( r, x ) : ( r − λ ) + x ≤ µ (cid:9) and, for λ > , we consider a smooth function u λ with cylindrical symmetrysuch that u λ ( r, x ) = s if ( r, x ) ∈ T λ,λ/ if ( r, x ) / ∈ T λ,λ/ (69)where s is such that W ( s ) s < inf V − sup V (see (W )). Moreover we mayassume that |∇ u λ ( r, x ) | ≤ r, x ) ∈ T λ,λ/ \ T λ,λ/ . (70)We have Λ( u λ ) = R h |∇ u λ | + ℓ u λ r + 2 V u i dx R u λ + R W ( u λ ) dx R u λ . (71)By (70) and (69) a direct computation shows that Z |∇ u λ | ≤ meas ( T λ,λ/ \ T λ,λ/ ) ≤ c λ (72) Z u λ r ≤ c λ meas ( T λ,λ/ ) ≤ c λ (73) Z u λ ≥ c meas ( T λ,λ/ ) ≥ c λ (74)where c , .., c are positive constants. So that R h |∇ u λ | + ℓ u λ r + 2 V u λ i dx R u λ ≤ sup V + O (cid:18) λ (cid:19) . (75)20ow Z T λ,λ/ \ T λ,λ/ | W ( u λ ) | ≤ c meas ( T λ,λ/ \ T λ,λ/ ) ≤ c λ Then Z W ( u λ ) dx = W ( s ) meas ( T λ,λ/ ) + Z T λ,λ/ \ T λ,λ/ W ( u λ ) ≤ W ( s ) meas ( T λ,λ/ ) + Z T λ,λ/ \ T λ,λ/ | W ( u λ ) |≤ W ( s ) meas ( T λ,λ/ ) + c λ . So R W ( u λ ) dx R u λ ≤ W ( s ) meas ( T λ,λ/ ) + c λ R u λ ≤ W ( s ) meas ( T λ,λ/ ) R u λ + c λ R u λ . (76)Now, since W ( s ) < , we have W ( s ) meas ( T λ,λ/ ) R u λ ≤ W ( s ) meas ( T λ,λ/ ) s meas ( T λ,λ/ ) = W ( s ) s (cid:18) λλ + 2 (cid:19) . (77)Then by (74), (76) and (77) we have R W ( u λ ) dx R u λ ≤ W ( s ) s (cid:18) λλ + 2 (cid:19) + c λ . (78)By (71), (75) and (78) we getΛ( u λ ) ≤ sup V + W ( s ) s (cid:18) λλ + 2 (cid:19) + O (cid:18) λ (cid:19) . sup V − inf V + inf V + W ( s ) s (cid:18) λλ + 2 (cid:19) + O (cid:18) λ (cid:19) . By lemma 17 we have lim inf u ∈ X r , k u k Lt → Λ( u ) ≥ inf V thenΛ( u λ ) ≤ sup V − inf V + lim inf u ∈ X r , k u k Lt → Λ( u ) + W ( s ) s (cid:18) λλ + 2 (cid:19) + O (cid:18) λ (cid:19) . (79)By assumption (W ) for λ large we havesup V − inf V + W ( s ) s (cid:18) λλ + 2 (cid:19) + O (cid:18) λ (cid:19) < . (80)21y (79) and (80) we get that for λ largeΛ( u λ ) < lim inf u ∈ X r , k u k Lt → Λ( u ) . (81)On the other hand, since by Lemma 18 kk L t satisfies the property (5), wehave by (6) that lim inf u ∈ X r , k u k Lt → Λ( u ) ≤ Λ . (82)Clearly (81) and (82) imply that assumption (3) is satisfied. (cid:3) Solitons are particular states of a dynamical system described by one or morepartial differential equations. Thus we assume that the states of this systemare described by one or more fields which mathematically are represented byfunctions u : R N → V (83)where V is a vector space with norm | · | V and which is called the internalparameters space. We assume the system to be deterministic; this means thatit can be described as a dynamical system ( X, γ ) where X is the set of the statesand γ : R × X → X is the time evolution map. If u ( x ) ∈ X, the evolution ofthe system will be described by the function u ( t, x ) := γ t u ( x ) . (84)We assume that the states of X have ”finite energy” so that they decay at ∞ sufficiently fast. Roughly speaking, the solitons are ”bump” solutions charac-terized by some form of stability.To define them at this level of abstractness, we need to recall some wellknown notions in the theory of dynamical systems. Definition 20
A set Γ ⊂ X is called invariant if ∀ u ∈ Γ , ∀ t ∈ R , γ t u ∈ Γ . Definition 21
Let ( X, d ) be a metric space and let ( X, γ ) be a dynamical sys-tem. An invariant set Γ ⊂ X is called stable, if ∀ ε > , ∃ δ > , ∀ u ∈ X , d ( u , Γ) ≤ δ, implies that ∀ t ≥ , d ( γ t u , Γ) ≤ ε. G be a subgroup of ( R N , +) and consider the following action T z of G on X : for all z ∈ G and u ∈ XT z u ( x ) = u ( x + z ) . Now we are ready to give the definition of soliton:
Definition 22
A state u ( x ) ∈ X is called soliton if there is an invariant set Γ such that • (i) ∀ t, γ t u ( x ) ∈ Γ , • (ii) Γ is stable, • (iii) Γ is G -compact (Def. 3) Remark 23
The above definition needs some explanation. For simplicity, weassume that Γ is a manifold (actually, in many concrete models, this is thegeneric case). Then (iii) implies that Γ is finite dimensional. Since Γ is invari-ant, u ∈ Γ ⇒ γ t u ∈ Γ for every time. Thus, since Γ is finite dimensional,the evolution of u is described by a finite number of parameters . The dynam-ical system (Γ , γ ) behaves as a point in a finite dimensional phase space. Bythe stability of Γ , a small perturbation of u remains close to Γ . However, inthis case, its evolution depends on an infinite number of parameters. Thus, thissystem appears as a finite dimensional system with a small perturbation.
Remark 24
The type of stability described above is called orbital stability inthe literature relative to the nonlinear Schr¨odinger and Klein-Gordon equations . We now assume that the dynamical system (
X, γ ) has two constants of motion.These constants can be considered as functionals on X. One of them will becalled energy and it will be denoted by E ; the other will be called hyleniccharge and it will be denoted by C .At this level of abstractness, the names energy and hylenic charge are con-ventional but E and C satisfy different assumptions; see assumption (EC-3) insection 2.1. In our applications to PDE’s, E will be the usual energy. The namehylenic charge has been introduced in [9], [4] and [5].The presence of E and C allows to give the following definition of hylomor-phic soliton. Definition 25
A soliton u ∈ X is called hylomorphic if Γ (as in Def. 22) hasthe following structure Γ = Γ ( e , c ) = { u ∈ X | E ( u ) = e , C ( u ) = c } where e = min { E ( u ) | C ( u ) = c } (85) for some c ∈ R . u satisfies thefollowing nonlinear eigenvalue problem: E ′ ( u ) = λC ′ ( u ) . Clearly, for a given c the minimum e in (85) might not exist; moreover,even if the minimum exists, it is possible that Γ does not satisfy (ii) or (iii) ofdef. 22.The following theorem holds Theorem 26
Assume that the dynamical system ( X, γ ) satisfies (EC-1),...,(EC-4) and (3). Moreover assume that E and C are two constants of motion.Then there exists ¯ δ > such that the dynamical system ( X, γ ) admits a family u δ ( δ ∈ (cid:0) , ¯ δ (cid:1) ) of hylomorphic solitons. The proof of this theorem will be given in the next section.
In order to prove Theorem 26 it is sufficient to show that the minimizers in Th.8 provide solitons, so we have to prove that the set Γ c δ is stable. To do this, weneed the (well known) Liapunov theorem in following form: Theorem 27
Let Γ be an invariant set and assume that there exists a differ-entiable function V (called a Liapunov function) such that • (a) V ( u ) ≥ and V ( u ) = 0 ⇔ u ∈ Γ • (b) ∂ t V ( γ t ( u )) ≤ • (c) V ( u n ) → ⇔ d ( u n , Γ) → . Then Γ is stable. Proof.
For completeness, we give a proof of this well known result. Arguingby contradiction, assume that Γ , satisfying the assumptions of Th. 27, is notstable. Then there exists ε > u n ∈ X and t n > d ( u n , Γ) → d ( γ t n ( u n ) , Γ) > ε. (86)Then we have d ( u n , Γ) → ⇒ V ( u n ) → ⇒ V ( γ t n ( u n )) → ⇒ d ( γ t n ( u n ) , Γ) → (cid:3) heorem 28 Assume (EC-1) and (EC-2). For u ∈ X and e , c ∈ R , we set V ( u ) = ( E ( u ) − e ) + ( C ( u ) − c ) . (87) If V is G -compact (see Def. 4) and Γ = { u ∈ X : E ( u ) = e , C ( u ) = c } 6 = ∅ , (88) then every u ∈ Γ is a soliton. Proof : We have to prove that Γ in (88) satisfies (i),(ii) and (iii) of Def. 22.The property (iii), namely the fact that Γ is G-compact, is a trivial consequenceof the fact that Γ is the set of minimizers of a G-compact functional V (seedefinitions 3 and 4). The invariance property (i) is clearly satisfied since E and C are constants of the motion. It remains to prove (ii), namely that Γ is stable.To this end we shall use Th. 27. So we need to show that V ( u ) satisfies (a), (b)and (c). Statements (a) and (b) are trivial. Now we prove (c). First we show theimplication ⇒ . Let u n be a sequence such that V ( u n ) → . By contradictionwe assume that d ( u n , Γ) , namely that there is a subsequence u ′ n such that d ( u ′ n , Γ) ≥ a > . (89)Since V ( u n ) → V ( u ′ n ) → , and, since V is G compact, there exists asequence g n in G such that, for a subsequence u ′′ n , we have g n u ′′ n → u . Then d ( u ′′ n , Γ) = d ( g n u ′′ n , Γ) ≤ d ( g n u ′′ n , u ) → ⇐ . Let u n be a sequence such that d ( u n , Γ) → , then there exists v n ∈ Γ s.t. d ( u n , Γ) ≥ d ( u n , v n ) − n . (90)Since V is G-compact, also Γ is G-compact; so, for a suitable sequence g n ,we have g n v n → ¯w ∈ Γ . We get the conclusion if we show that V ( u n ) → . Wehave by (90), that d ( u n , v n ) → d ( g n u n , g n v n ) → g n v n → ¯w , we have g n u n → ¯w ∈ Γ . Therefore, by the continuity of V and since ¯w ∈ Γ , we have V ( g n u n ) → V ( ¯w ) = 0 and we can conclude that V ( u n ) → . (cid:3) In the cases in which we are interested, X is an infinite dimensional manifold;then if you choose generic e and c , V is not G -compact since the set Γ = { u ∈ X : E ( u ) = e , C ( u ) = c } has codimension 2. However, Th. (8) allowsto determine e and c in such a way that V is G -compact and hence to provethe existence of solitons by using Theorem 28. Proof of Th. 26.
In order to prove Th. 26, we will use Th. 28 with e = e δ and c = c δ where e δ and c δ are given by Th. 8.25e set V ( u ) = ( E ( u ) − e δ ) + ( C ( u ) − c δ ) . (91)We show that V is G -compact: let w n be a minimizing sequence for V, then V ( w n ) → E ( w n ) → e δ and C ( w n ) → c δ . Let J δ be as inTheorem 8. Now, since inf J δ = e δ c δ + δ [ e δ + ac sδ ] , we have that w n is a minimizing sequence also for J δ . Then, since J δ is G -compact, we get w n is G -compact . (92)So we conclude that V is G -compact and hence the conclusion follows by usingTheorem 28. (cid:3) In this section we shall study the existence of hylomorphic solitons on latticefor the Schr¨odinger equation (NSE) in R N . The existence of solitons for (NSE) is an old problem and there are manyresults in the case V = 0 ([20], [19], [6] and the references in [9]).Here we assume that V is a lattice potential, namely we assume that thepotential V satisfies the periodicity condition. V ( x ) = V ( x + Az ) for all x ∈ R N and z ∈ Z N ( V ′ )where A is a N × N invertible matrix.Here we look for solitons and do not require they to be vortices, so the energycorresponds to the expression (34) with ℓ = 0 , namely E ( u ) = E ( u ) = Z R (cid:20) |∇ u | + V ( x ) u + W ( u ) (cid:21) dx, u ∈ X. (93)As before the charge is C ( u ) = Z u dx. In this case X is the ordinary H ( R N ) Sobolev space. We shall consider thefollowing action of the group G = Z N on X :for all z ∈ Z N and u ∈ X : T z u ( x ) = u ( x + Az ) . (94)Clearly the charge C is G -invariant and, since V satisfies ( V ′ ), also the energy E is invariant under this group action. The following Theorem holds:26 heorem 29 Let W and V satisfy assumptions (W ) , ..., ( W ) and (V ) , (V ′ ) . Then there exists ¯ δ > such that the dynamical system described by the Schr¨odingerequation (NSE) has a family u δ ( δ ∈ (cid:0) , ¯ δ (cid:1) ) of hylomorphic solitons. The proof of this theorem is based on the abstract theorem 26. In this casethe energy is given by (93). We need to show that assumptions ( W ) , ..., ( W )and ( V ) , ( V ′ ) permit to show that assumptions (EC-1), ...,(EC-4) and (3) oftheorem 26 are satisfied.(EC-1) and (EC-2) are trivially verified. The proof of the other assumptionsfollows the same lines of the proof of Th. 12 as we can see in the followinglemmas: Lemma 30 E and C satisfy the coercivity assumption (EC-3). Proof.
The proof is the same of that of lemma 15 with ℓ = 0. (cid:3) Lemma 31 E and C satisfy the splitting property (EC-4). Proof.
The proof is the same of that of lemma 16 with ℓ = 0. (cid:3) Lemma 32 If < t < NN − , N ≥ , the norm k u k L t satisfies the property (5),namely { u n is a vanishing sequence } ⇒ k u n k L t → . Proof.
We set for j ∈ Z N Q j = A (cid:0) j + Q (cid:1) = (cid:8) Aj + Aq : q ∈ Q (cid:9) where Q is now the cube defined as follows Q = (cid:8) ( x , .., x n ) ∈ R N : 0 ≤ x i < (cid:9) .Now let x ∈ R N and set y = A − ( x ) . Clearly there exist q ∈ Q and j ∈ Z N such that y = j + q. So x = Ay = A ( j + q ) ∈ Q j . Then we conclude that R N = [ j Q j . Let u n be a bounded sequence in H (cid:0) R N (cid:1) such that, up to a subsequence, k u n k L t ≥ a > . We need to show that u n is non vanishing. Then, if C is the27onstant for the Sobolev embedding H ( Q j ) ⊂ L t ( Q j ) and k u n k H ≤ M, wehave 0 < a t ≤ Z | u n | t = X j Z Q j | u n | t = X j k u n k t − L t ( Q j ) k u n k L t ( Q j ) ≤ (cid:18) sup j k u n k t − L t ( Q j ) (cid:19) · X j k u n k L t ( Q j ) ≤ C (cid:18) sup j k u n k t − L t ( Q j ) (cid:19) · X j k u n k H ( Q j ) = C (cid:18) sup j k u n k t − L t ( Q j ) (cid:19) k u n k H ≤ CM (cid:18) sup j k u n k t − L t ( Q j ) (cid:19) . Then (cid:18) sup j k u n k L t ( Q j ) (cid:19) ≥ (cid:18) a t CM (cid:19) / ( t − Then, for any n, there exists j n ∈ Z N such that k u n k L t ( Q jn ) ≥ α > . (95)Then, if we set Q = AQ , we easily have k T j n u n k L t ( Q ) = k u n k L t ( Q jn ) ≥ α > . (96)Since u n is bounded, also T j n u n is bounded in H ( R N ) . Then we have, upto a subsequence, that T j n u n ⇀ u weakly in H ( R N ) and hence strongly in L t ( Q ). By (96), u = 0 . (cid:3) Lemma 33
Assumption (3) is satisfied namely inf u ∈ H ( R N ) Λ( u ) < Λ Proof.
This lemma is analogous to lemma 19, however in this case the proofis easier: since X = H ( R N ) , we need only to construct a function u ∈ H ( R N )such that Λ( u ) < Λ . Such a function can be constructed as follows. Set u R = s if | x | < R if | x | > R + 1 | x | R s − ( | x | − R ) R +1 R s if R < | x | < R + 1 . Then Z |∇ u R | dx = O ( R N − ) , Z | u R | dx = O ( R N ) , so that 28 h |∇ u R | + 2 V u R i dx R u R ≤ sup V + O (cid:18) R (cid:19) . (97)Moreover Z W ( u R ) dx = W ( s ) meas ( B R ) + Z B R +1 \ B R W ( u R ) . So R W ( u R ) dx R u R ≤ W ( s ) meas ( B R ) + c R N − R u λ ≤ ( since W ( s ) < ≤ W ( s ) meas ( B R ) s meas ( B R +1 ) + c R N − R N = W ( s ) s (cid:18) RR + 1 (cid:19) N + c R . (98)Then, by (97) e (98) we getΛ( u R ) ≤ sup V + W ( s ) s (cid:18) RR + 1 (cid:19) N + O (cid:18) R (cid:19) . By lemma 17 we have inf V ≤ lim inf k u k Lt → Λ( u ) , thenΛ( u R ) ≤ lim inf k u k Lt → Λ( u ) + sup V − inf V + W ( s ) s (cid:18) RR + 1 (cid:19) N + O (cid:18) R (cid:19) . (99)On the other hand, since by Lemma 18 the L t norm satisfies the property(5), we have by (6) that lim inf k u k Lt → Λ( u ) ≤ Λ . (100)Clearly (99), (100) and assumption (W ) imply that for R large we haveΛ( u R ) < Λ . Then assumption (3) is satisfied. (cid:3)
Proof of Th. 29 . The proof follows from Th. 26 and Lemmas 30, 31 and33. (cid:3)
In this section we shall apply the abstract theorem 26 to the existence of hylo-morphic solitons in R N for the nonlinear Klein-Gordon equation (NKG). Thereare well known results on the existence of stable solutions for (NKG) ([27], [23])29nd more recently the existence of hylomorphic solitons for (NKG) has beenstudied in [4] and 2.More exactly, we consider the equation (cid:3) ψ + W ′ ( ψ ) = 0 (NKG)where (cid:3) = ∂ t − ∆, ψ : R × R N → C ( N ≥
3) , W : C → R and W ′ are as in(27) (see the beginning of section 3.1). Assume that W ( s ) = 12 m s + N ( s ) , s ≥ , m = 0 (101)where N ( s ) = o ( s ) . We make the following assumptions on W : • (NKG-i) (Positivity ) W ( s ) ≥ s ≥ • (NKG-ii) (Hylomorphy ) ∃ s ∈ R + such that W ( s ) < m s • (NKG-iii) (Growth condition ) there are constants c , c > , < r, q < N/ ( N −
2) such that for any s > | N ′ ( s ) | ≤ c s r − + c s q − . We shall assume that the initial value problem is well posed for (NKG). Eq.(NKG) is the Euler-Lagrange equation of the action functional S ( ψ ) = Z (cid:18) | ∂ t ψ | − |∇ ψ | − W ( ψ ) (cid:19) dxdt. (102)The energy and the charge take the following form: E ( ψ ) = Z (cid:20) | ∂ t ψ | + 12 |∇ ψ | + W ( ψ ) (cid:21) dx (103) C ( ψ ) = − Re Z i∂ t ψψ dx. (104)(the sign ”minus”in front of the integral is a useful convention). We set X = H ( R N , C ) × L ( R N , C )and we will denote the generic element of X by u = ( ψ ( x ) , ˆ ψ ( x )); then, by thewell posedness assumption, for every u ∈ X, there is a unique solution ψ ( t, x )of (NKG) such that ψ (0 , x ) = ψ ( x ) (105) ∂ t ψ (0 , x ) = ˆ ψ ( x ) . ∂ t ψ = ˆ ψ (106) ∂ t ˆ ψ = ∆ ψ − W ′ ( ψ ) . (107)The time evolution map γ : R × X → X is defined by γ t u ( x ) = u ( t, x )where u ( x ) = ( ψ ( x ) , ˆ ψ ( x )) ∈ X and u ( t, x ) = ( ψ ( t, x ) , ˆ ψ ( t, x )) is the uniquesolution of (106) and (107) satisfying the initial conditions (105). The energyand the charge, as functionals defined in X, become E ( u ) = Z (cid:20) (cid:12)(cid:12)(cid:12) ˆ ψ (cid:12)(cid:12)(cid:12) + 12 |∇ ψ | + W ( ψ ) (cid:21) dx (108) C ( u ) = − Re Z i ˆ ψψ dx. (109) The following Theorem holds:
Theorem 34
Assume that W satisfies (NKG-i),...,(NKG-iii). Then there ex-ists ¯ δ > such that the dynamical system described by the equation (NKG) hasa family u δ ( δ ∈ (cid:0) , ¯ δ (cid:1) ) of hylomorphic solitons. The proof of this theorem is based on the abstract theorem 26. In thiscase the energy E and the hylenic charge C have the form (103) and (104)respectively.Assumption (EC-1) is clearly satisfied. E and C are invariant under trans-lations, so assumption (EC-2) is satisfied with respect to the action T z of thegroup G = R N where T z u ( x ) = u ( x + z ) , z ∈ R N . It can be seen that the coercitivity assumption (EC-3) is satisfied with a = 0 . Arguing as in lemma 16 (replacing W by N ) it can be shown that also (EC-4)is satisfied, namely that E and C satisfy the splitting property.It remains to prove (3). First of all we set: k u k ♯ = (cid:13)(cid:13)(cid:13) ( ψ, ˆ ψ ) (cid:13)(cid:13)(cid:13) ♯ = max ( k ψ k L r , k ψ k L q )where r, q are introduced in (NKG-iii). With some abuse of notation we shallwrite max ( k ψ k L r , k ψ k L q ) = k ψ k ♯ . 31 emma 35 The norm k u k ♯ satisfies the property (5), namely { u n is a vanishing sequence } ⇒ k ψ n k ♯ → . Proof.
Let ψ n be a bounded sequence in H (cid:0) R N (cid:1) such that, up to asubsequence, k ψ n k ♯ ≥ a > . We need to show that ψ n is non vanishing. Maybe taking a subsequence, we have that at least one of the following holds: • (i) k ψ n k ♯ = k ψ n k L r • (ii) k ψ n k ♯ = k ψ n k L q Suppose that (i) holds. Then, we argue as il lemma 32. If (ii) holds, weargue in the same way replacing r with q. (cid:3) Now we setΛ ♯ = lim inf k u k ♯ → Λ( u ) = lim ε → inf n Λ( ψ, ˆ ψ ) | ˆ ψ ∈ L ; ψ ∈ H ; k ψ k ♯ < ε o . By remark 9, we have that Λ ≥ Λ ♯ ; so let us evaluate Λ ♯ . Lemma 36 If W satisfies assumption (NKG-iii), then the following inequalityholds Λ ♯ ≥ m. Proof.
By (NKG-iii) we have (cid:12)(cid:12)(cid:12)(cid:12)Z N ( | ψ | ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k Z | ψ | r + k Z | ψ | q ≤ k k ψ k r♯ + k k ψ k q♯ . If we assume that k ψ k ♯ = 1, (cid:12)(cid:12)(cid:12)(cid:12)Z N ( | εψ | ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ε r + k ε q . Now, choose s ∈ (2 , min( r, q )) . Thus, if ε > r, q > s > ε s Z (cid:16) |∇ ψ | + m | ψ | (cid:17) dx − (cid:12)(cid:12)(cid:12)(cid:12)Z N ( | εψ | ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≥ ε s k k ψ k ♯ − k ε r − k ε q = k ε s − k ε r − k ε q ≥ (cid:12)(cid:12)(cid:12)(cid:12)Z N ( ε | ψ | ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε s Z (cid:16) |∇ ψ | + m | ψ | (cid:17) dx. (110)Now we clearly have 32 ♯ = lim ε → inf n Λ( εψ, ˆ ψ ) | ˆ ψ ∈ L , ψ ∈ H , k ψ k ♯ = 1 o . (111)Let us estimate Λ( εψ, ˆ ψ ) using (110):Λ( εψ, ˆ ψ ) = R (cid:18)(cid:12)(cid:12)(cid:12) ˆ ψ (cid:12)(cid:12)(cid:12) + |∇ εψ | + m | εψ | (cid:19) dx + R N ( | εψ | ) dx (cid:12)(cid:12)(cid:12) Re R i ˆ ψεψ dx (cid:12)(cid:12)(cid:12) ≥ R (cid:12)(cid:12)(cid:12) ˆ ψ (cid:12)(cid:12)(cid:12) + (cid:16) ε − ε s (cid:17) R (cid:16) |∇ ψ | + m | ψ | (cid:17) ε (cid:12)(cid:12)(cid:12) Re R i ˆ ψψ dx (cid:12)(cid:12)(cid:12) ≥ R (cid:12)(cid:12)(cid:12) ˆ ψ (cid:12)(cid:12)(cid:12) + ε (cid:0) − ε s − (cid:1) m R | ψ | ε (cid:18)R (cid:12)(cid:12)(cid:12) ˆ ψ (cid:12)(cid:12)(cid:12) dx (cid:19) / (cid:16)R | ψ | dx (cid:17) / ≥ (cid:18)R (cid:12)(cid:12)(cid:12) ˆ ψ (cid:12)(cid:12)(cid:12) dx (cid:19) / · εm √ − ε s − (cid:16)R | ψ | dx (cid:17) / ε (cid:18)R (cid:12)(cid:12)(cid:12) ˆ ψ (cid:12)(cid:12)(cid:12) dx (cid:19) / (cid:16)R | ψ | dx (cid:17) / = m p − ε s − . Then lim Λ( εψ, ˆ ψ ) ≥ m. (112)So the conclusion follows by (111) and (112). (cid:3) Next we will show that the hylomorphy assumption (3) is satisfied.
Lemma 37
Assume that W satisfies (NKG-i),...,(NKG-iii), then inf u ∈ X Λ( u ) < Λ . Proof . Let
R >
0; set u R = s if | x | < R if | x | > R + 1 | x | R s − ( | x | − R ) R +1 R s if R < | x | < R + 1 . (113)By (NKG-ii) there exists 0 < β < m such that W ( s ) ≤ β s . (114)We set ψ = u R , and ˆ ψ = βu R . u ∈ X Λ( u ) = inf ψ, ˆ ψ R (cid:18) (cid:12)(cid:12)(cid:12) ˆ ψ (cid:12)(cid:12)(cid:12) + |∇ ψ | + W ( ψ ) (cid:19) dx (cid:12)(cid:12)(cid:12) Re R i ˆ ψψ dx (cid:12)(cid:12)(cid:12) ≤ R (cid:16) β | u R | + |∇ u R | + W ( u R ) (cid:17) dxβ R | u R | dx ≤ R | x |
The assumptions (EC-1),...,(EC-4) and (3) are satisfied,then the proof follows by using Th. 26. (cid:3)
We conclude this section with the following theorem which gives some moreinformation on the structure of the solitons:
Theorem 38
Let u be a hylomorphic soliton relative to the equation (NKG)with initial data u ( x ) = ( ψ ( x ) , ˆ ψ ( x )) ∈ X . Then there exists ω ∈ R such that ψ satisfies the equation − ∆ ψ + W ′ ( ψ ) = ω ψ , (116)ˆ ψ = − iωψ and γ t u ( x ) = (cid:20) ψ ( x ) e − iωt − iωψ ( x ) e − iωt (cid:21) . (117)34 roof. Since u is a hylomorphic soliton it is a critical point of E constrainedon the manifold M c = { u ∈ X : C ( u ) = c } . Clearly E ′ ( u ) = − ωC ′ ( u ) (118)where − ω is a Lagrange multiplier. We now compute the derivatives E ′ ( u ) , C ′ ( u ) . For all ( v , v ) ∈ X = H ( R N , C ) × L ( R N , C ) , we have E ′ ( u ) (cid:20) v v (cid:21) = Re Z h ˆ ψ v + ∇ ψ ∇ v + W ′ ( x, ψ ) v i dxC ′ ( u ) (cid:20) v v (cid:21) = − Re Z (cid:16) i ˆ ψ v + iv ψ (cid:17) dx = − Re Z (cid:16) i ˆ ψ v + iv ψ (cid:17) dx = − Re Z (cid:16) i ˆ ψ v − iψ v (cid:17) dx. Then (118) can be written as follows:Re
Z (cid:2) ∇ ψ ∇ v + W ′ ( x, ψ ) v (cid:3) dx = ω Re Z i ˆ ψ v dx Re Z ˆ ψ v dx = − ω Re Z iψ v dx. Then − ∆ ψ + W ′ ( x, ψ ) = iω ˆ ψ ˆ ψ = − iωψ . (119)So we get (116). From (116) and (119), we easily verify that (117) solves(106), (107). (cid:3) References [1]
Abrikosov A.A.,
On the magnetic properties of superconductors of thesecond group,
Sov. Phys. JETP (1957), 1174-1182.[2] Badiale M. , Benci V. , Rolando S. , A nonlinear elliptic equation withsingular potential and applications to nonlinear field equations , J. Eur.Math. Soc. (2007), 355-381.[3] Badiale M., Benci V., Rolando S.,
Three dimensional vortices in thenonlinear wave equation,
Bollettino U.M.I., Serie 9, II (2009), 105-134.354]
Bellazzini J., Benci V., Bonanno C., Micheletti A.M.,
Solitonsfor the Nonlinear Klein-Gordon-Equation , Advances in Nonlinear Studies, (2010), 481-500 (arXiv:0712.1103).[5]
Bellazzini J., Benci V., Bonanno C., Sinibaldi E.,
Hylomorphicsolitons in the nonlinear Klein-Gordon equation,
Dynamics of Partial Dif-ferential Equations, (2009), 311-333. (arXiv:0810.5079).[6] Bellazzini J., Benci V., Ghimenti M., Micheletti A.M.,
On theexistence of the fundamental eigenvalue of an elliptic problem in R N , Adv.Nonlinear Stud. (2007), 439–458.[7] Bellazzini J., Bonanno C.,
Nonlinear Schr¨odinger equations withstrongly singular potentials.
Proceedings of the Royal Society of Edinburgh:Section A Mathematics 140: (2010), 707-721.[8]
Bellazzini J., Bonanno C., Siciliano G.,
Magnetostatic vortices intwo dimensional Abelian gauge theory, Mediterranean J. Math.,
Mediter-ranean Journal of Mathematics, (2009), 347–366 .[9] Benci V,
Hylomorphic solitons,
Milan J. Math. (2009), 271-332.[10] Benci V. Fortunato D.,
Three dimensional vortices in Abelian GaugeTheories,
Nonlinear Analysis T.M.A., (2009), 4402-4421.[11] Benci V., Fortunato D.,
Hylomorphic Solitons on lattices,
Discrete andcontinuous dynamical systems,
28 (
Benci V., Fortunato D.,
On the existence of stable charged Q-balls,
Jour-nal of Mathematical Physics, to appear. (ArXive. 1011.5044).[13]
Benci V. Fortunato D.,
Hamiltonian formulation of the Klein-Gordom-Maxwell equations,
Rend. Lincei Mat. Appl. (2011), 1-22..[14] Benci V., Fortunato D.,
Existence of solitons in nonlinear beam equa-tion, (
ArXive 1102.5315).[15]
Benci V. Fortunato D.,
Existence of hylomorphic solitary waves inKlein-Gordon and in Klein-Gordon-Maxwell equations,
Rend. Lincei Mat.Appl. (2009), 243-279. (arXiv:0903.3508).[16] Benci V. Fortunato D.,
Spinning Q-balls for the Klein-Gordon-Maxwellequations,
Comm. Math. Phys., (2010), 639-668.[17]
Benci V, Ghimenti M., Micheletti A.M.,
The Nonlinear Schroedingerequation: solitons dynamics , Journal of Differential Equations , (2010),3312-3341 (arXiv:0812.4152).[18]
Benci V. , Visciglia N. , Solitary waves with non vanishing angular mo-mentum , Adv. Nonlinear Stud. (2003), 151-160.3619] Buslaev V. S., Sulem C.,
On asymptotic stability of solitary waves fornonlinear Schr¨odinger equations.
Annales de l’institut Henri Poincar´e (C)Analyse non lin´eaire, (2003), 419-475.[20] Cazenave T., Lions P.L. ,
Orbital stability of standing waves for somenonlinear Schr¨odinger equations , Comm. Math. Phys. (1982), no. 4,549–561.[21] Esteban M., Lions P.L.
A compactness lemma,
Nonlinear Analysis, (1983), 381-385.[22] Gelfand I.M., Fomin S.V.,
Calculus of Variations,
Englewood Cliffs,NJ. Prentice-Hall, (1963).[23]
Grillakis M., Shatah J,, Strauss W.,
Stability theory of solitary wavesin the presence of symmetry, I , J. Funct. Anal. (1987), 160–197.[24] Kim C. , Kim S. , Kim Y.
Global nontopological vortices , Phys. Review D, ,(1985), 5434-5443.[25] Nielsen H., Olesen P.,
Vortex-line models for dual strings,
Nucl. Phys.B , (1973), 45-61.[26] Palais R.S.,
The principle of symmetric criticality,
Comm. Math. Phys., , (1979), 19-30.[27] Shatah J. ,
Stable Standing waves of Nonlinear Klein-Gordon Equations,
Comm. Math. Phys., , (1983), 313-327.[28] Vilenkin A., Shellard E.P.S.,
Cosmic strings and other topologicaldefects,
Cambrige University press, Cambridge (1994).[29]
Volkov M.S.,
Existence of spinning solitons in field theory, arXiv:hep-th/0401030 (2004)[30]
Volkov M.S. , W¨ohnert E. , Spinning Q -balls , Phys. Rev. D66