Math. Ann. (2018) 371:707–740 https://doi.org/10.1007/s00208-018-1666-z

Mathematische Annalen

Long time dynamics for semi-relativistic NLS and half wave in arbitrary dimension Jacopo Bellazzini1 · Vladimir Georgiev2,3,4 · Nicola Visciglia2

Received: 1 September 2017 / Revised: 11 February 2018 / Published online: 10 March 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract We consider the Cauchy problems associated with semi-relativistic NLS (sNLS) and half wave (HW). In particular we focus on the following two main questions: local/global Cauchy theory; existence and stability/instability of ground states. In between other results, we prove the existence and stability of ground states for sNLS in the L 2 supercritical regime. This is in sharp contrast with the instability of ground states for the corresponding HW, which is also established along the paper, by showing an inflation of norms phenomenon. Concerning the Cauchy theory we show, under radial symmetry assumption the following results: a local existence result in H 1

Communicated by Y. Giga. V.G. is supported in part by INDAM, GNAMPA–Gruppo Nazionale per l’Analisi Matematica, la Probabilita e le loro Applicazion, by Institute of Mathematics and Informatics, Bulgarian Academy of Sciences and by Top Global University Project, Waseda University. J.B. and V.G. are supported by Gnampa 2017 project “Problemi stazionari e di evoluzioni per equazioni di campo non lineari”.

B

Nicola Visciglia [email protected] Jacopo Bellazzini [email protected] Vladimir Georgiev [email protected]

1

Università di Sassari, Via Piandanna 4, 70100 Sassari, Italy

2

Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56100 Pisa, Italy

3

Faculty of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan

4

IMI–BAS, Acad. Georgi Bonchev Str., Block 8, 1113 Sofia, Bulgaria

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for energy subcritical nonlinearity and a global existence result in the L 2 subcritical regime.

1 Introduction The aim of this paper is the analysis of the following Cauchy problems with special emphasis to the local/global existence and uniqueness results, as well as to the issue of existence and stability/instability of ground states: 

i∂t u = Au − u|u| p−1 , (t, x) ∈ R × Rn u(0, x) = f (x) ∈ H s (Rn ),

(1.1)

√ √ where A = − and A = 1 − , namely Half Wave (HW) and semi-relativistic NLS (sNLS). Since now on H s (Rn ) and H˙ s (Rn ) denote respectively the usual inhomogeneous and homogeneous Sobolev spaces in Rn , endowed with the norms s (Rn ) as to the (1 − )s/2 u L 2 (Rn ) and (−)s/2 u L 2 (Rn ) . We shall also refer to Hrad s n set of functions belonging to H (R ) which are radially symmetric. Along the paper we shall study several properties of the Cauchy problems associated with sNLS and HW. The first result will concern the local/global Cauchy theory at low regularity under an extra radiality assumption. We point out that at the best of our knowledge in the literature there exist very few results about the global existence of solutions to both HW and sNLS (see e.g. [20]). In particular we mention the result in [22] where it is considered HW in dimension n = 1 with nonlinearity u|u|3 and initial data in H 1 (R), without any further symmetry assumption. Indeed the aforementioned result can be extended to sNLS for n = 1 with quartic nonlinearity. We also underline that in one space dimension no results are available concerning the global existence for higher order nonlinearity, namely p > 4. One novelty in this paper is that we provide global existence results in higher dimension n ≥ 2 under the radial symmetry assumption, provided that p satisfies some restrictions. Finally we recall the paper [18] for a general local Cauchy theory for fractional dispersive equations, but not including HW and sNLS equations and [16] for Cauchy theory for cubic HW in one space dimension. Another important issue considered along this article is the existence and stability/instability properties of solitary waves associated with sNLS and HW. Of course, the first main ingredient in order to speak about dynamical properties of the solitary waves, is a robust Cauchy theory that at the best of our knowledge is provided in this paper for the first time in the radially symmetric setting. We recall that two values of the nonlinearity p are quite relevant: the nonlinearity u|u|2/(n−1) , which is H 1/2 (Rn )-critical, and the nonlinearity u|u|2/n , which is L 2 (Rn )critical. Next we present our main result about the Cauchy problems (1.1): we prove on 1 (Rn ) via contraction argument for H 1/2 (Rn ) one hand a local existence result in Hrad subcritical nonlinearity; on the other hand we show that the solutions are global in time provided that the nonlinearity is L 2 (Rn )-subcritical. √ √ 2 Theorem 1.1 Let n ≥ 2, A be either − or 1 − , p ∈ (1, 1 + n−1 ). Then for every R > 0 there exists T = T (R) > 0 and a Banach space X T such that:

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1 (Rn )); • X T ⊂ C([0, T ]; Hrad 1 (Rn ) with  f  • for any f (x) ∈ Hrad H 1 (Rn ) ≤ R, there exists a unique solution u(t, x) ∈ X T of (1.1).

Assume moreover that p ∈ (1, 1 + n2 ), then the solution is global in time. We point out that by a cheap argument (based only on Sobolev embedding and energy estimates) one can solve locally in time the Cauchy problem (1.1) for initial data f (x) ∈ H n/2+ (Rn ) (without any radiality assumption). Notice that we provide a local existence result, in radial symmetry, with regularity H 1 (Rn ) for n ≥ 2. Indeed it will be clear to the reader, by looking at the proof of Theorem 1.1, that one can push 1/2+ the local theory at the level of regularity Hrad (Rn ). The main technical difficulty 1/2+ 1 (Rn ) to H n to go from Hrad rad (R ) being the fact that in the first case we work with straight derivatives, and hence the weighted chain rules that we need along the proof are straightforward. In the second case the proof requires more delicate commutator estimates that we prefer to skip along this paper. We also underline that in the L 2 subcritical regime we get a global existence result. Next we shall analyze the issue of standing waves. We recall that standing waves are special solutions to (1.1) with a special structure, namely u(t, x) = eiωt v(x), where ω ∈ R plays the role of the frequency. Indeed u(t, x) is a standing wave solution if and only if v(x) satisfies Av + ωv − v|v| p = 0

in Rn .

(1.2)

It is worth mentioning that, following the pioneering paper [7], it is well understood how to build up solitary waves for both sNLS and HW via a energy constrained minimization argument, in the case of L 2 -subcritical nonlinearity. Moreover as a byproduct of this variational approach, the corresponding solitary waves are orbitally stable. In the sequel the following quantities, preserved respectively by sNLS and HW, will play a crucial role: 1 1 p+1 u L p+1 (Rn ) , (1.3) Es (u) = u2H 1/2 (Rn ) − 2 p+1 (namely the energy associated with sNLS) and 1 Ehw (u) = E˜hw (u) + u22 , 2

(1.4)

1 where E˜hw (u) = 21 u2H˙ 1/2 (Rn ) − p+1 u L p+1 (Rn ) is the energy associated with HW. Moreover we recall the conservation of the mass, i.e. p+1

d u(t, x)2L 2 (Rn ) = 0 dt

(1.5)

for solutions u(t, x) associated with (1.1). In the nonlocal context in which we are interested in, the minimization problems analogue of the one studied in [7] for NLS are the following ones:

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Jrs = inf Es (u),

Jrhw = inf Ehw (u)

(1.6)

  1/2 n 2 Sr = u ∈ H (R ) s.t. u L 2 (Rn ) = r .

(1.7)

u∈Sr

where

u∈Sr

Indeed it is not difficult (following the rather classical concentration-compactness argument, see for instance [2] for more details in the non-local setting) to get a strong compactness property (up to translation) for minimizing sequences associated with the minimization problems above, provided that the nonlinearity is L 2 subcritical, i.e. 1 < p < 1 + n2 . By combining this fact with the global existence result stated in Theorem 1.1, one can prove a stability result that we state below. In order to do that first we need to introduce a suitable notion of stability, that is weaker respect to the usual one. This is due mainly to the fact that we are not able to get any global existence result for the Cauchy problem associated with sNLS and HW at the level of regularity of the Hamiltonian H 1/2 and without the radiality assumption. Hence we need to assume more regularity and also the radial symmetry on the perturbations allowed along the definition of stability. 1 (Rn ) be bounded in H 1/2 (Rn ). We say that N is weakly Definition 1.1 Let N ⊂ Hrad orbitally stable by the flow associated with sNLS (resp. HW) if for any  > 0 there exists δ > 0 such that 1 (Rn ) dist H 1/2 (u(0, .), N ) < δ and u(0, x) ∈ Hrad ⇒ t (u(0, .)) is globally defined and sup dist H 1/2 (t (u(0, .)), N ) <  t

where dist H 1/2 denotes the usual distance with respect to the topology of H 1/2 and t (u(0, .) is the unique global solution associated with the Cauchy problem sNLS (resp. HW) and with initial condition u(0, x). We can now state the next result, where we use the notations (1.6) and (1.7). We state it as a corollary since it is a classical consequence of the concentration-compactness argument in the spirit of [7] and Theorem 1.1, that guarantees a global dynamic for sNLS and HW. Hence we shall not provide the straightforward proof along the paper. Nevertheless we believe that it has its own interest. Corollary 1.1 Let 1 < p < 1 +

2 n

and n ≥ 1. Then for every r > 0 we have:

• Jrs > −∞ (resp. Jrhw > −∞) and Brs = ∅ (resp. Brhw = ∅) where   Brs := v ∈ Sr s.t. Es (v) = Jrs (resp. Brhw := {v ∈ Sr s.t. Ehw (v) = Jrhw }). In particular for every v ∈ Brs (resp. v ∈ Brhw ) there exists ω ∈ R such that √ 1 − v + ωv − v|v| p−1 = 0 (resp.

123

√ −v + ωv − v|v| p−1 = 0);

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• the set Brs (resp. Brhw ) is weakly orbitally stable by the flow associated with sNLS (resp. HW). Moreover in the case n = 1 the weak orbital stability property can 1 (R) by be strengthened, in the sense that in the Definition 1.1 we can replace Hrad 1 the larger space H (R). On the contrary, the situation dramatically changes in the L 2 -supercritical regime (namely p > 1 + n2 ) since the aforementioned minimization problems (1.6) are meaningless, in the sense that: Jrs = Jrhw = −∞, ∀r > 0,

p >1+

2 . n

(1.8)

Next result is aimed to show a special geometry (local minima) for the constrained 2 energy associated to sNLS in the L 2 -supercritical regime, i.e 1 + n2 < p < 1 + n−1 . In order to state our next result let us first introduce a family of localized and constrained minimization problems: (1.9) Jr = inf Es (u), u∈Sr ∩B1

where 1

Bρ = {u ∈ H 1/2 (Rn ) s.t. (1 − ) 4 u L 2 (Rn ) ≤ ρ}. We also recall that the notion of weak orbital stability is given in Definition 1.1. 2 and n ≥ 1. There exists r0 > 0 such that the Theorem 1.2 Let 1 + n2 < p < 1 + n−1 following conditions occur for every r ∈ (0, r0 ):

• Jr > −∞, Br = ∅ and Br ⊂ B1/2 ∩ H 1 (Rn ), where Br := {v ∈ Sr ∩ B1 s.t. Es (v) = Jr } . In particular for every v ∈ Br there exists ω ∈ R such that √

1 − v + ωv − v|v| p−1 = 0;

• the elements in Br are ground states on Sr , namely:   inf Es (w) = Jr where Cr = w ∈ Sr s.t. Es | Sr (w) = 0 . Cr

Assume moreover the following assumption: sup

(−T− ( f ),T+ ( f ))

u(t, x) H 1/2 (Rn ) < ∞ ⇒ T± ( f ) = ∞

(1.10)

where (−T− ( f ), T+ ( f )) is the maximal time of existence of u(t, x) which is the 1 (Rn ). Then we get: nonlinear solution to sNLS with initial datum f (x) ∈ Hrad 1/2

• the set Br ∩ Hrad (Rn ) is weakly orbitally stable for the flow associated with sNLS.

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We point out that the extra assumption (1.10) it is satisfied for n = 1 and p = 4 without any radiality assumption (it follows by a suitable adaptation to sNLS of the argument given in [22] for HW). An alternative and simpler argument for the global existence of one dimensional quartic sNLS is given in the Appendix. Hence the statement above provides the existence of stable standing waves for the quartic sNLS for n = 1, by removing the condition (1.10). More precisely we can state the following result. Corollary 1.2 Let n = 1 and p = 4. Then under the same notations as in Theorem 1.2 we have that for r < r0 the corresponding set Br is weakly orbitally stable. Indeed Br satisfies a straightened version of the property given in Definition 1.1, where we can 1 (R) by H 1 (R). replace Hrad We point out that the weak orbital stability stated in Theorem 1.2 under the condition (1.10), as well as in Corollary 1.2, is a byproduct of the general Cazenave–Lions strategy (see [7]), once the following compactness property (where we do not assume any radiality assumption) is established: u k ∈ Sr ∩ B1 , Es (u k ) → Jr ⇒ ∃xk ∈ Rn s.t. u k (x + xk ) has a strong limit in H 1/2 (Rn ). The main difficulty here being the fact that we have to deal with a local minimization problems [since the global minimization problem is meaningless, see (1.8)] and hence the application of the concentration-compactness argument is much more delicate. We also underline that if we look at the same minimization problems as above, under the extra radiality assumption (namely u k (x) = u k (|x|), then the compactness stated above occurs without any selection of the translation parameters xk . We would like to mention that at the best of our knowledge stable solitary waves are proved to exist in the L 2 -supercritical regime only in presence of an external confining potential, see e.g. [3]. Theorem 1.2 is the first result about translation invariant equations. In order to state our last result about existence/instability of ground states for HW, we need to introduce also the following functional: n( p − 1) 1 p+1 u2H˙ 1/2 (Rn ) − u L p+1 (Rn ) , 2 2( p + 1)

P(u) = and the corresponding set:

  1/2 n M = u ∈ H (R ) s.t. P(u) = 0 .

(1.11)

It is well known (see [27]) that we have the following inclusion    w ∈ Sr s.t. Ehw | Sr = 0 ⊂ M, namely every critical point of the energy Ehw on the constraint Sr belongs to the set M. It is worth mentioning that this fact is reminiscent of the Pohozaev identity, which

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is here adapted to the case of HW. The following minimization problem will be crucial in the sequel: Ir = inf Ehw (u). Sr ∩M

Theorem 1.3 Let n ≥ 2 and 1 +

2 n

< p <1+

2 n−1 .

Then for every r > 0 we have:

• Ir > −∞ and Ar = ∅, where Ar := {v ∈ Sr ∩ M s.t. Ehw (v) = Ir } . Moreover any v ∈ Ar satisfies √ −v + ωv − v|v| p−1 = 0 for a suitable ω ∈ R; 1 (Rn ) satisfies E ( f ) < I and P( f ) < 0, n ≥ 2 • assume f (x) ∈ Sr ∩ Hrad hw r √ and u(t, x) is solution to (1.1) (where A = −), then the following alternative holds: either the solution blows-up in finite time or u(t, x) H˙ 1/2 (Rn ) ≥ eat for suitable a > 0. In particular the set Ar is not weakly orbitally stable for the flow associated with HW. Notice that in the first part of the statement, which is mostly variational, we don’t assume the radial symmetry. On the contrary in the statement about the evolution along the Cauchy problem we assume the radiality. This is mainly due to the fact that at the best of our knowledge no global Cauchy theory is available without the radiality assumption. We shall emphasize that the existence of positive ground states is an already know fact in all space dimensions by minimizing a suitable Weinstein-like functional. We underline that positive ground states are unique up to space shift and phase multiplication, see [12,13]. Here, in the first part of the statement, we present a different proof of the existence of ground states that involves the functional P. Our variational approach is useful to characterize the dynamics near the ground states. We restrict to the case n ≥ 2 just because we focus on the dynamics near the ground state that is available only when n ≥ 2. We also underline that our approach to prove the second part of Theorem 1.3, namely the norm inflation, is inspired by the work of Ogawa–Tsutsumi [21] that was based, in the context of the classical NLS, on the analysis of time derivative of the localized virial  ¯ R · ∇ud x Mϕ R (u) = 2 Im u∇ϕ where ϕ R is a rescaled cut-off function such that ∇ϕ R (x) ≡ x for |x| ≤ R and ∇ϕ R (x) ≡ 0 for |x| >> R. This approach has been further extended by Boulenger– Himmelsbach–Lenzmann [4] in the non-local context with dispersion (−)s for 21 < s < 1. In this paper we shall take advantage of similar computations in the case of HW.

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Fig. 1 Qualitative behavior of the constrained energy functionals associated with sNLS and HW. In this qualitative picture the kinetic energy is given by u H˙ 1/2 (Rn )

We point out that the discrepancy between the dynamics for sNLS and HW, revealed by Theorems 1.2 and 1.3 about the stability/instability of ground states (namely Br and Ar ) in the L 2 supercritical regime, is reminiscent of the results of [14,15] for the dispersive equation describing a Boson Star: i∂t u =



m2

− u −

1 2

|u| u, (t, x) ∈ R × R3 . |x|

Indeed, in [14] it has been proved that ground states are unstable by blow up if m = 0, while in [15] it is shown that the ground states are orbitally stable whenever m > 0. However our situation is rather different from the one describing a Boson Star, in fact in our case the constrained energy functional is always unbounded from below for any assigned L 2 constraint. We shall also mention [9,10] concerning global Cauchy theory for a general class of Hartree nonlinearity (Fig. 1). We conclude with a picture showing the main difference between the functionals Ehw and Es revealed by Theorems 1.2 and 1.3, in the L 2 supercritical regime. In fact in the first case we have established the stability of the ground states, and in the second case we have proved on the contrary its instability. For the HW (upper curve in the figure) the functional Ehw admits a critical point of mountain pass type. For sNLS (lower curve in the figure) we have the existence of a local minimizer for Es .

2 The Cauchy theory for HW and sNLS The aim of this section it to prove a local/global existence and uniqueness result for the Cauchy problem (1.1). We need several tools that we shall exploit along the proof. We treat in some details the result for the HW, and we say at the end how to transfer the results at the level of sNLS. As usual we shall look for fixed point of the integral operator associated with the Cauchy problem for HW:

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S f (u) = e−it

√

−



t

f +i

e−i(t−τ )

√

−

715

u(τ )|u(τ )| p−1 dτ.

(2.1)

0 1 (Rn ). We perform a fixed point argument in a suitable space where f (x) ∈ Hrad 1 (Rn )) that, as we shall see below, is provided by an interpolation X T ⊂ C([0, T ]; Hrad between a Kato-smoothing type estimate and the usual energy estimates. 1/2+

2.1 A Brezis–Gallouët–Strauss Type Inequality in Hr ad

(Rn )

In this subsection we introduce two functional inequalities that will be useful respectively to achieve the local Cauchy theory and the globalization argument, following in the spirit the paper by Strauss (see [30]) and Brezis–Gallouët (see [6]). Proposition 2.1 For every n ≥ 2 and s > 1/2 there exists a constant C = C(s, n) > 0 such that:  n−1    (2.2) |x| 2 u  ∞ n ≤ Cu H s (Rn ) ; L (R )  

 n−1 

u H s (Rn )    , (2.3) |x| 2 u  ∞ n ≤ Cu H 1/2 (Rn ) ln 2 + L (R ) u H 1/2 (Rn ) s (Rn ). for every u ∈ Hrad

Proof In the radial case it is well-known the Strauss estimate [30] |x|

n−1 2

1 |u(x)| ≤ C f  H 1 (Rn ) , ∀u ∈ Hrad (Rn )

that has been extended for fractional Sobolev space in [8]. Moreover in [28,29] it has been proved that |x|

n−1 2

1/2

|u j (x)| ≤ Cu j  H 1/2 (Rn ) , ∀u ∈ Hrad (Rn ), ∀ j ≥ 0.

(2.4)

Here we use the notation u j = ϕj

√  − u, ∀ j ≥ 0,

and ϕ j (s) is the usual Paley–Littlewood decomposition, namely ϕ j (s) ∈ C0∞ ((0, ∞)) are non-negative function supported in [2 j−1 , 2 j+1 ], such that: 

ϕ j (s) = 1, ∀s ≥ 0.

j≥0

The homogeneous variant of the Strauss inequality was established in [8] n

s |x| 2 −s |u(x)| ≤ C f  H˙ s (Rn ) , ∀u ∈ Hrad (Rn ),

1 n
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However, in this work the inhomogeneous estimate (2.4) is sufficient.  Notice that the first estimate (2.2) follows by decomposing u(x) = j≥0 u j (x) and by noticing that by Minkowski inequality and (2.4)  n−1    |x| 2 u 

L ∞ (Rn )

≤

   n−1  |x| 2 u j  j

L ∞ (Rn )

≤C



u j  H 1/2 (Rn ) ≤ Cu H s (Rn )

j

where at the last step we have used the Cauchy–Schwartz inequality and the assumption s > 1/2. Concerning the proof of (2.3) we refine the argument above as follows:  n−1    |x| 2 u 

L ∞ (Rn )

≤C

∞ 

u j  H 1/2 (Rn ) =

j=0

M  j=0



∞ 

u j  H 1/2 (Rn ) + 



S1 (M)

j=M+1



u j  H 1/2 (Rn ) 

S2 (M)



with M being sufficiently large integer. We can estimate these two terms by Cauchy– Schwartz as follows: √ S1 (M) ≤ C Mu H 1/2 (Rn ) , S2 (M) ≤ C2−M(1/4+s/2)) u H s (Rn ) so we get   n−1   |x| 2 u(x)

∞

√ ≤ C Mu H 1/2 (Rn ) + C2−M(1/4+s/2) u H s (Rn ) .

We conclude by choosing: 

u H s (Rn ) M = ln 2 + u H 1/2 (Rn )

 .  

2.2 Energy estimates and Kato smoothing Next proposition is the key estimate for the linear propagator, that will suggest the space X T where to perform a fixed point argument. In the sequel we shall use the notation [x]δ = |x|1+δ + |x|1−δ . Proposition 2.2 Let δ > 0 be fixed. We have the following bound −1

[x]δ q eit

123

√

−

f  L q (R;L 2 (Rn )) ≤ C f  L 2 (Rn )

Long time dynamics for semi-relativistic NLS and half wave…

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for every q ∈ [2, ∞] and C > 0 is an universal constant that does not depend on q. Proof By interpolation it is sufficient to treat q = 2, ∞. The case q = ∞ is trivial √ it − f  L 2 (Rn ) =  f  L 2 (Rn ) . and follows by the isometry e The case q = 2 follows by combining the Kato smoothing (see [19,24]) together with the following √ lemma that provides uniform weighted estimates for the resolvent associated with −. Lemma 2.1 Let δ > 0 be fixed, then we have the following uniform bounds:    −1   − 21 √   21  [x]   − − (λ + i) f ≤ C [x]δ f   δ  L 2 (Rn )

L 2 (Rn )

where C > 0 does not depend on λ,  > 0. Proof We have the following identity √

− − (λ + i)

−1

=

√

−1   − + λ + i ◦ − − (λ + i)2

and hence the desired estimate follows by the following well-known estimates available for the resolvent associated with the Laplacian operator − (see [1,25]):      − 21 √   21  2 −1  [x] [x] f  −(− − (λ + i) ) f ≤ C  δ  2 n  δ  2 n L (R )

L (R )

and    − 12  [x] (− − (λ + i)2 )−1 f   δ 

L 2 (Rn )

≤

  21 C [x] |λ + i|  δ

  f 

L 2 (Rn )

.  

The proof of Proposition 2.2 is complete. Next we present a-priori estimates associated with the Duhamel operator. Proposition 2.3 Let δ > 0 be fixed. For every q1 ∈ [2, ∞] and q2 ∈ (2, ∞] we get      − q11 t i(t−s)√−  [x] e F(s)ds   δ 0

L q1 (R;L 2 (Rn ))

   q12   [x] ≤C F  δ 



L q2 (R;L 2 (Rn ))

.

Proof The proof follows by combining Proposition 2.2 with the T T ∗ argument (see [17]) in conjunction with the Christ–Kiselev Lemma (see [11]). More precisely let T be the following operator: T : L 2 (Rn )  f → eit

√

−

f ∈ Xq

where G(t, x) X q

   − q1   = [x]δ G(t, x) 

L q (R;L 2 (Rn ))

.

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Notice that T is continuous by Proposition 2.2, and hence by a duality argument we get the continuity of the operator  √ ∗ e−is − F(s)ds ∈ L 2 (Rn ). T : Yq  F(t, x) → R

1/q

(here we have used the dual norm G(t, x)Yq = [x]δ G(t, x) L q  (R;L 2 (Rn )) .) As a consequence, by choosing respectively q = q1 and q = q2 in the estimates above, we deduce that the following operator is continuous:  √ ∗ T ◦ T : Yq2  F(t, x) → ei(t−s) − F(s)ds ∈ X q1 R

and hence

  √    ei(t−s) − F(s)ds    R

X q1

≤ CFYq2 .

In fact by a straightforward localization argument (namely choose F(s, x) supported only for s > 0) we get  ∞  √   i(t−s) −   e F(s)ds ≤ CFYq2 .   0

X q1

Notice that this estimate looks very much like  t except  ∞ the one that we want to prove, that we would like to replace the integral 0 by the truncated integral 0 . This is possible thanks to the general Christ–Kiselev Lemma mentioned above, that works   provided that q2 < q1 . 2.3 Local Cauchy theory in Hr1ad (Rd ) We define the space X T and we perform in X T a contraction argument for the integral operator S f (see (2.1)). We introduce q, q¯ > 2 and δ > 0 such that −

−1 + δ (n − 1)( p − 1) 1 − δ + = 2 q¯ q

(2.5)

where u|u| p−1 is the nonlinearity. 2 ). Next we introduce Notice that q, q, ¯ δ as above exist provided that p ∈ (1, 1+ n−1 the space X T whose norm is defined as    − q1   u X T = u(t, x) L ∞ H 1 (Rn ) + [x]δ ∇x u(t, x)  T

   − q1   [x] + u(t, x)  δ 

123

q

L T L 2 (Rn )

q

L T L 2 (Rn )

(2.6)

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(here and below we use he notation L rT (X ) = L r ((0, T ); X )). Next we introduce a cut-off function ψ ∈ Cc∞ (Rn ) with ψ(x) = 0 for |x| > 2 and ψ(x) = 1 for |x| < 1 and we write the forcing term u|u| p−1 = ψu|u| p−1 + (1 − ψ)u|u| p−1 . ¯ Then we have, by using Propositions 2.2 and 2.3 where we choose (q1 , q2 ) = (∞, q) ¯ and where we apply the operator ∇x (that commutes with the and (q1 , q2 ) = (q, q) equation): S f u X T

 1    q¯  p−1   ≤ C f  H 1 (Rn ) + C [x]δ ∇x ψ(u|u| )   q¯ L T L 2 (Rn )         + C ∇x ((1 − ψ)(u|u| p−1 )) 1 2 n + C u|u| p−1  L T L (R )

L 1T L 2 (Rn )

where S f is the integral equation defined in (2.1). Then we get by using the Leibnitz rule and the properties of the cut-off function ψ:    − (n−1)( p−1) 1−δ  n−1  p−1   q¯ 2 2 |u| |x| |x| S f u X T ≤ C f  H 1 + C  |x| ∇ u x      − (n−1)( p−1) −1+δ  n−1  p−1   q¯ 2 2 |u| |x| |x| +C |x| u   q¯  2 L T L (|x|<2)    p−1  p + C |u| ∇x u  1 2 + C T u L ∞ L 2 p (Rn ) . L T L (|x|>1)

q¯ 

L T L 2 (|x|<2)

(2.7)

T

n−1

Next notice that by (2.5), by the estimate |x| 2 u L ∞ (Rn ) ≤ Cu H 1 (Rn ) for every 1 (Rn ), by Hölder in space and time, and by the Sobolev embedding H 1 (Rn ) ⊂ u ∈ Hrad 2 p L (Rn )) we get: · · · ≤ C f  H 1 (Rn ) + C T

1− q1¯ − q1

p−1

p

u L ∞ H 1 (Rn ) u X T + C T u L ∞ H 1 (Rn ) . T

T

From this estimate one can conclude that S f : B X T (0, R) → B X T (0, R) for suitable T, R > 0 where   B X T (0, R) = v(t, x) ∈ X T s.t. v X T ≤ R . Next we endow the set B X T (0, R) with the following distance:    − q1   [x] u(t, x) d(u 1 , u 2 ) = u 1 − u 2  L ∞ L 2 (Rn ) +   δ  T

q

L T L 2 (Rn )

.

It is easy to check that the metric space (B X T (0, R), d) is complete. Then we conclude provided that we show that the map S f is a contraction on this space. In order to do

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that we notice that by using the estimates in Proposition 2.3 (but we don’t apply in this case the operator ∇x ) then we get:    − q1   [x] (S u − S u ) S f u 1 (t) − S f u 2 (t) L ∞ L 2 (Rm ) +  f 1 f 2   δ T

≤ CT

1− q1¯ − q1

q

   −1  p−1 p−1  q  u 1  X T + u 1  X T [x]δ (u 1 − u 2 ) 

  p−1 p−1 + C T u 1  X T + u 1  X T u 1 − u 2  L ∞ L 2 (Rm )

L T L 2 (Rn )

q

L T L 2 (Rn )

T

Hence by choosing T > 0 small enough and by recalling that u 1 , u 2 ∈ B X T (0, R) then we get: d(S f u 1 , S f u 2 ) ≤

1 d(u 1 , u 2 ), ∀u 1 , u 2 ∈ B X T (0, R). 2

We conclude by using the contraction mapping principle. 2.4 Conditional global existence in Hr1ad (Rn ) for 1 < p < 1 +

2 n

First notice that for 1 < p < 1 + n2 , then from Gagliardo–Nirenberg inequality we get sup

(−T− ( f ),T+ ( f ))

u(t, x) H 1/2 (Rn ) < ∞

(2.8)

where (−T− ( f ), T+ ( f )) is the maximal time of existence. By arguing as in the Sect. 2.3 (and by using the fact that u(t, x) is a solution) we get:   n−1  p−1    − (n−1)( p−1) 1−δ  q¯ 2 |x| ∇ |x| u |x| 2 |u| u X T ≤ C f  H 1 (Rn ) + C  x  

q¯ 

L T L 2 (|x|<2)

+ C|u| p−1 ∇x u L 1 L 2 (|x|>1) . T

By (2.5) and Hölder in time we get: u X T ≤ C f  H 1 (Rn )    1−δ  + C |x| q ∇x u 

q¯ 

L T L 2 (|x|<2)

   n−1  p−1 + T u L ∞ H 1 (Rn ) |x| 2 u  ∞ ∞ T

L T L (Rn )

   n−1  p−1 1− q1¯ − q1 ≤ C f  H 1 (Rn ) + C T u X T +T u L ∞ H 1 (Rn ) |x| 2 u  ∞ ∞

L T L (Rn )

T

Hence if we introduce the function g(T ) = supt∈(0,T ) u X t we obtain: g(T ) ≤ C f  H 1 (Rn ) + C max{T, T

123

1− q1¯ − q1

}g(T )|x|

n−1 2

p−1

u L ∞ L ∞ (Rn ) . T

.

Long time dynamics for semi-relativistic NLS and half wave…

721

By using (2.3) in conjunction with the assumption supt u(t, x) H 1/2 (Rn ) < ∞, we have: g(T ) ≤ C f  H 1 (Rn ) + C max{T, T

1− q1¯ − q1

}g(T ) ln

p−1 2

(2 + Cg(T )).

(2.9)

Next we prove, as consequence of the estimate above, the following: p−1 1− 1 − 1 Claim Let T¯ > 0 be s. t. C max{T¯ , T¯ q¯ q } ln 2 (2 + 2C 2  f  H 1 (Rn ) ) = 21 then g(T¯ ) ≤ 2C f  H 1 (Rn ) . In order to prove the claim notice that if it is not true then there exists T˜ < T¯ such that g(T˜ ) = 2C f  H 1 (Rn ) . Then by going back to the proof of (2.9) and by using the property T˜ < T¯ , one can prove: 1

1

1− − g(T˜ ) ≤ C f  H 1 (Rn ) + C max{T˜ , T˜ q¯ q }g(T˜ ) ln

< C f  H 1 (Rn ) + C max{T¯ , T¯

1− q1¯ − q1

}g(T˜ ) ln

p−1 2 p−1 2



 2 + Cg(T˜ )   2 + 2C 2  f  H 1 (Rn )

1 = C f  H 1 (Rn ) + g(T˜ ) 2 and then we get g(T˜ ) < 2C f  H 1 (Rn ) , hence contradicting the definition of T˜ . By an iteration argument (based on the claim above) we can construct a sequence T¯ j such that 1− q1¯ − q1

C max{T¯ j , T¯ j

} ln

p−1 2

  1 2 + 2C 2 u(T j ) H 1 (Rn ) = , 2

(2.10)

and g(T j+1 ) ≤ 2 j+1 C j+1  f  H 1 (Rn ) ,

(2.11)

where T j+1 = T¯1 + · · · + T¯ j . We claim that T j → ∞ as j → ∞, and in this case we conclude. In fact if this is the case then the solution can be extended to the interval [0, T j ] for every j > 0 and of course it implies global well-posedness since T j → ∞. Of course if there is a subsequence T jk ≥ 1 then we conclude, and hence it is not restrictive to assume T j < 1 at least for large j. In particular we get 1− q1¯ − q1

T¯ j



1− q1¯ − q1

= max T¯ j , T¯ j





 p−1   − ( p−1) 2 ∼ O ln u(T j ) H 1 (Rn ) ≥ C j− 2

(2.12) where we used (2.10) and the fact that (2.11) implies u(T j ) H 1 (Rn ) ≤ 2 j C j  f  H 1 (Rn ) . We conclude since by (2.5) we can choose (q, q) ¯ such that q1¯ + q1 = (n−1)(2 p−1) + 0 with 0 > 0 arbitrarily small. In fact it implies, together with (2.12), the following estimate: T¯ j0 +1 =

j0  j=1

T¯ j ≥ C

j0 

j

p−1 − (2−(n−1)( p−1)−2

0)

j=1

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and the r.h.s. is divergent (for small 0 > 0) provided that p <1+

p−1 2−(n−1)( p−1)

< 1, namely

2 n.

2.5 Cauchy theory for sNLS The idea is to reduce the Cauchy theory for sNLS to the Cauchy theory for HW that has been √ √ established above. More precisely let us introduce the operator L = 1 −  − − and hence we can rewrite the Cauchy problem associated with sNLS as follows:  √ i∂t u + − = −Lu + u|u| p−1 , (t, x) ∈ R × Rn , (2.13) 1 (Rn ). u(0, x) = f (x) ∈ Hrad Notice that the operator L corresponds in Fourier at the multiplier √

1 , 1+|ξ |2 +|ξ |

and

hence we have L : H s (Rn ) → H s (Rn ). Thanks to this property it is easy to check that we can perform a fixed point argument for (2.13) in the space X T following the same argument used to solve above for HW. The minor change concerns the fact that the extra term Lu is absorbed in the nonlinear perturbation. Also the globalization argument given in the Sect. 2.4 can be easily adapted to sNLS.

3 Existence and stability of solitary waves for sNLS The main point along the proof of Theorem 1.2 is the proof of the compactness (up to translation) of the minimizing sequences associated with Jr , as well as the proof of the fact that the minimizers belong to B1/2 ∩ Sr , provided that r is small enough. This is sufficient in order to deduce that the minimizers are far away from the boundary and hence are constrained critical points. In particular they satisfy the Euler–Lagrange equation up to the Lagrange multiplier ω. Another delicate issue is to show that the local minimizers (namely the elements in Br according with the notation in Theorem 1.2) have indeed minimal energy between all the critical points of Es constrained on the whole sphere Sr , provided that r > 0 is small. Next we shall focus on the points above, and we split the proofs in several steps. We also mention that the statement about the orbital stability it follows easily by the classical argument of Cazenave–Lions (see [7]) once a nice Cauchy theory has been established. 3.1 Local minima structure We start with the following lemma that shows a local minima structure for the functional Es on the constraint Sr , for r small enough. Proposition 3.1 There exists r0 > 0 such that: inf

{u∈Sr |u H 1/2 (Rn ) =1}

123

Es (u) >

1 , ∀r < r0 ; 4

(3.1)

Long time dynamics for semi-relativistic NLS and half wave…

inf √ Es (u) {u∈Sr |u H 1/2 (Rn ) ≤2 r }

<

723

r , ∀r < r0 . 2

(3.2)

Proof By the Gagliardo–Nirenberg inequality we get for some 0 > 0: 1 1 1 p+1 0 u2H 1/2 (Rn ) − u L p+1 (Rn ) ≥ u2H 1/2 (Rn ) − C0 r γ0 u2+ H 1/2 (Rn ) 2 p+1 2   1 = u2H 1/2 (Rn ) 1 − C0 r γ0 uH0 1/2 (Rn ) , ∀u ∈ Sr (3.3) 2 and hence 1 1 1 p+1 u2H 1/2 (Rn ) − u L p+1 (Rn ) > , ∀u ∈ Sr , u H 1/2 (Rn ) = 1. 2 p+1 4 Concerning the bound (3.2) notice that: 1 1 p+1 u2H 1/2 (Rn ) − u L p+1 (Rn ) 2 p+1 1 1 1 1 p+1 = u2H 1/2 (Rn ) − u2L 2 (Rn ) + u2L 2 (Rn ) − u L p+1 (Rn ) 2 2 2 p+1 and by using Plancharel 1 1 r 1 p+1 u2H 1/2 (Rn ) − u2L 2 (Rn ) + − u L p+1 (Rn ) 2 2 2 p+1 1 r 1 p+1 = u2H + − u L p+1 (Rn ) , ∀u ∈ H 1/2 (Rn ) ∩ Sr 2 2 p+1

··· =

where  u2H =

Rn

|ξ |2  |u(ξ ˆ )|2 dξ. 1 + 1 + |ξ |2

In particular we get u2H ≤

1 u2H˙ 1 (Rn ) , ∀u ∈ H˙ 1 (Rn ) s.t. u(ξ ˆ ) = 0, ∀|ξ | > 1. 2

(3.4)

Next we fix ϕ smooth, such that: ϕ(ξ ˆ ) = 0, ∀|ξ | ≥ 1 and ϕ2L 2 (Rn ) = r . We also

introduce ϕλ (x) where ϕˆ λ (ξ ) = λn/2 ϕ(λξ ˆ ), then we get by the inequalities above [next we restrict to λ > 1 in order to guarantee ϕˆλ (ξ ) = 0, ∀|ξ | > 1 and hence we can apply (3.4)]:

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1 1 p+1 ϕλ 2H 1/2 (Rn ) − ϕλ  L p+1 (Rn ) 2 p+1 1 1 r p+1 ϕλ  L p+1 (Rn ) . ≤ ϕλ 2H˙ 1 (Rn ) + − 2 2 p+1 Notice that ϕλ ∈ Sr . Moreover by a rescaling argument we get 1 1 p+1 ϕλ 2H˙ 1 (Rn ) − ϕλ  L p+1 (Rn ) < 0 2 p+1 for any λ large enough. We conclude since  ϕλ 2H 1/2 (Rn ) =

Rn

 |ϕ| ˆ 2 1+

|ξ |2 dξ → ϕ2L 2 (Rn ) = r, λ2

as λ → ∞

√ and hence for λ large enough ϕλ  H 1/2 (Rn ) < 2 r .

 

3.2 Avoiding vanishing Next result will be crucial to exclude vanishing for the minimizing sequences. In the sequel r0 > 0 is the number that appears in Proposition 3.1. Proposition 3.2 Assume r < r0 and u k ∈ Sr ∩ B1 be such that Es (u k ) → Jr then lim inf k→∞ u k  L p+1 (Rn ) > 0. Proof Assume by the absurd that it is false. Then we get by Proposition 3.1 1 r > Jr = lim Es (u k ) = lim u k 2H 1/2 (Rn ) . k→∞ k→∞ 2 2 This is a contradiction since u k ∈ Sr and hence u k 2H 1/2 (Rn ) ≥ r .

 

3.3 Avoiding dichotomy Next result will be crucial to avoid dichotomy. Proposition 3.3 There exists r1 > 0 such that for any 0 < r < l < r1 we have r Jl < lJr . We shall need the following result that allows us to get a bound on the size of the minimizing sequences. Lemma 3.1 There exists r2 > 0 such that inf √ Es (u) {u∈Sr |u H 1/2 (Rn ) ≤2 r }

123

<

Es (u), √ inf {u∈Sr |2 r ≤u H 1/2 (Rn ) <1}

∀r < r2 .

Long time dynamics for semi-relativistic NLS and half wave…

725

Proof In view of Proposition 3.1 it is sufficient to prove that √ inf {u∈Sr |2 r ≤u H 1/2 (Rn ) <1}

1 1 p+1 u2H 1/2 (Rn ) − u L p+1 (Rn ) 2 p+1

>

r . 2

In order to prove it, we go back to (3.3) and we get 1 1 p+1 u2H 1/2 (Rn ) − u L p+1 (Rn ) 2 p+1   1 ≥ u2H 1/2 (Rn ) 1 − C0 r γ0 uH0 1/2 (Rn ) , ∀u ∈ Sr 2 and hence √ · · · ≥ 2r (1 − C0 r γ0 ), ∀u ∈ Sr , 2 r ≤ u H 1/2 (Rn ) < 1.  

We conclude provided that r is small enough.

We can now conclude the proof of Proposition 3.3. Fix vk ∈ Sr , vk  H 1/2 (Rn ) ≤ 1 such that limk→∞ Es (vk ) = Jr . √ Notice that by Lemma 3.1 we can assume vk  H 1/2 (Rn ) < 2 r . In particular we have   √ l l vk ∈ Sl and vk  H 1/2 (Rn ) < 2 l, r r and hence  Jl ≤ lim inf Es k

l vk r



p+1

1l 1 l 2 p+1 = vk 2H 1/2 (Rn ) − vk  L p+1 (Rn ) . 2r p + 1 r p+1 2 Recall that by Proposition 3.2 we can assume vk  L p+1 (Rn ) > δ0 > 0 and hence we can continue the estimate above as follows  p+1 

2 1 l l 1 1 p+1 p+1 vk 2H 1/2 (Rn ) − vk  L p+1 (Rn ) + − p+1 vk  p+1 2 p+1 p+1 r r 2   p+1  p+1  l l δ0 δ0 l 2 l l 2 l l − p+1 ≤ Jr + − p+1 < Jr . = Es (vk ) + r r p + 1 r r p + 1 r r 2 r 2

l ··· = r



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3.4 Conclusions Notice that by Proposition 3.1 we deduce that the minimizers (if exist) have to belong necessarily to B1/2 . Next we prove the compactness, up to translations, of the minimizing sequences. Since now on we shall fix r small enough according with the Propositions above. Let u k ∈ Sr be such that u k  H 1/2 (Rn ) ≤ 1 and Es (u k ) → Jr , then by combining Proposition 3.2 and with the Lieb translation Lemma in H 1/2 (Rn ) (see [2]), we have that up to translation the weak limit of u k is u¯ = 0. Our aim is to prove that u¯ is a ¯ 2L 2 (Rn ) = r¯ then it is sufficient to prove strong limit in L 2 (Rn ). Hence if we denote u r¯ = r . Notice that we have by weak convergence u k − u ¯ 2L 2 (Rn ) + u ¯ 2L 2 (Rn ) = r + o(1) and if we assume (by subsequence) u k − u ¯ 2L 2 (Rn ) → t then we have t + r¯ = r . We shall prove that necessarily t = 0 and hence r = r¯ . Next by classical arguments, namely Brezis–Lieb Lemma (see [5]) and the Hilbert structure of H 1/2 (Rn ), we get Es (u k ) = Es (u k − u) ¯ + Es (u) ¯ + o(1) ≥ Ju k −u ¯ 2

L 2 (Rn )

+ Ju ¯ 2

L 2 (Rn )

+ o(1).

Passing to the limit as k → ∞, and by recalling t + r¯ = r we get Jt+¯r = Jr ≥ Jr¯ + Jt .

(3.5)

On the other hand by Proposition 3.3 we get (t + r¯ )Jr¯ > r¯ Jt+¯r and (t + r¯ )Jt > tJt+¯r that imply Jr¯ + Jt > Jt+¯r and it is in contradiction with (3.5). As a last step we have to prove that the local minima (namely the elements in Br ) minimize the energy Es among all the critical points of Es constrained to Sr , provided that r > 0 is small enough. In order to prove this fact we shall prove the following property: ∃r0 > 0 s.t. ∀r < r0 the following occurs w H 1/2 (Rn ) < 1/2, ∀w ∈ Sr s.t. Es | Sr = 0, Es (w) < Jr . Once this fact is established then we can conclude easily since it implies that if w ∈ Sr is a constraint critical points with energy below Jr and r < r0 , then w can be used as test functions to estimate Jr from above and we get Jr ≤ Es (w), hence we have a contradiction. In order to prove the property stated above recall that if w ∈ Sr is a critical point of Es restricted to Sr , then notice that it has to satisfy the following Pohozaev type identity (this follows by an adaptation of the argument in [27] to sNLS):

123

Long time dynamics for semi-relativistic NLS and half wave…

1 2

Q(w) = 0 where Q(w) =

 Rn



|ξ |2 1 + |ξ |2

|w| ˆ 2 dξ −

727

n( p − 1) p+1 w L p+1 (Rn ) 2( p + 1)

and hence 2 Q(w) n( p − 1)  |ξ |2 1 1 2  |w| ˆ 2 dξ = w H 1/2 (Rn ) − 2 n( p − 1) Rn 1 + |ξ |2 np − n − 2 ≥ w2H 1/2 (Rn ) . 2n( p − 1)

Es (w) = Es (w) −

Notice that np − n − 2 > 0 if p > 1 + n2 . From the estimate above we get np − n − 2 r w2H˙ 1/2 (Rn ) ≤ Es (w) < Jr < 2n( p − 1) 2 where we used Proposition 3.1 at the last step. It is now easy to conclude.

4 Existence/instability of solitary waves for HW This section is devoted to the proof of Theorem 1.3. 4.1 Existence of minimizer Even if Theorem 1.3 is stated for the energy Ehw (u) on Sr we shall work at the beginning on the unconstrained functional. At the end we shall come back to the constraint minimization problem as stated in Theorem 1.3. The first result concerns the fact that the constraint M (see (1.11)) is a natural constraint. Lemma 4.1 Let v ∈ H 1/2 (Rn ) be such that P(v) = 0, Ehw (v) = inf Ehw (u) u∈M

then √ −v + v − v p = 0 Proof We notice that since v is minimizer then

√ √ n( p − 1) p v −v − −v + v − v p = λ 2

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We claim that λ = 0. Notice that by the equation above we get p+1

v2H˙ 1/2 (Rn ) + v22 − v L p+1 (Rn ) = λv2H˙ 1/2 (Rn ) −

dλ( p − 1) p+1 v L p+1 (Rn ) (4.1) 2

Moreover since P(v) = 0 we get v2H˙ 1/2 (Rn ) =

n( p − 1) p+1 v L p+1 (Rn ) . ( p + 1)

(4.2)

Next notice that we have the following rescaling invariance 1

P(u) = 0 ⇒ P(μ p−1 u(μx) = 0 and hence since v is a minimizer we get mentary computations gives 1 2

1

d p−1 v(μx)) |μ=1 dμ Ehw (μ

= 0 that by ele-







p+1 2 1 2 − n v H˙ 1/2 (Rn ) + − n v2L 2 (Rn ) p−1 2 p−1

1 p+1 p+1 −n v L p+1 (Rn ) = 0. − p−1 p+1

By combining (4.1), (4.2), (4.3) we get easily λ = 0.

(4.3)  

The next result concerns the proof of the existence of a minimizer for the unconstrained problem. Lemma 4.2 Let 1 +

2 n

< p <1+

2 n−1 .

1/2

There exists w ∈ Hrad (Rn ) such that

P(w) = 0, Ehw (w) = inf Ehw (u). u∈M

Proof For simplicity we denote inf u∈M Ehw (u) := E0 . Notice that we have

1 n( p − 1) 1 p+1 − u L p+1 (Rn ) + u2L 2 (Rn ) , Ehw (u) = 2( p + 1) p+1 2 ∀u ∈ H 1/2 (Rn ), P(u) = 0

(4.4)

p−1) 1 2 and hence, since ( n( 2( p+1) − p+1 ) > 0 for p > 1 + n we get E0 ≥ 0. Let wk ∈ M be a minimizing sequence. We shall prove first the compactness of minimizing sequences by assuming radial 1/2 symmetry, namely wk ∈ Hrad (Rn ) such that 1/2

Ehw (wk ) → E0 , wk ∈ Hrad (Rn ), P(wk ) = 0.

123

(4.5)

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729

In a second step we shall prove that it is not restrictive to assume that wn can be assumed radially symmetric. First of all notice that we get supn wk  H 1/2 (Rn ) < ∞. In fact by using the constraint P(wk ) = 0 it is sufficient to check that sup wk  L 2 (Rn ) + wk  L p+1 (Rn ) < ∞

(4.6)

k

and it follows by the expression (4.4) of the energy on the constraint P(u) = 0. p+1 The next step is to show that inf k wk  L p+1 (Rn ) > 0. It follows by the following chain of inequalities wk 2H 1/2 (Rn ) =

n( p − 1) p+1 wk  L p+1 (Rn ) p+1 γ

0 0 ≤ Cwk  L02 (Rn ) wk 2+ ≤ Cwk 2+ L p+1 (Rn ) H 1/2 (Rn )

where we used the Gagliardo–Nirenberg inequality, the boundedness of wk  L 2 (Rn ) (see (4.6)) and the Sobolev embedding H 1/2 (Rn ) ⊂ L p+1 (Rn ). As a consequence we get inf k wk 2H 1/2 (Rn ) > 0 and since P(wk ) = 0 the same lower bound occurs for p+1

inf k wk  L p+1 (Rn ) > 0.

Next we introduce w¯ ∈ H 1/2 (Rn ) as the weak limit of wk . We are done if we show that the convergence is strong. First of all notice that by the compactness of the 1/2 Sobolev embedding Hrad (Rn ) → L p+1 (Rn ), we deduce wk → w¯ in L p+1 (Rn ) and p+1 hence w¯ = 0 since inf n wk  L p+1 (Rn ) > 0. Next notice that since P(wk ) = 0 then 1 n( p − 1) p+1 w ¯ 2H˙ 1/2 (Rn ) − w ¯ L p+1 (Rn ) ≤ 0. 2 2( p + 1) It implies by a continuity argument ∃λ¯ ∈ (0, 1]

s.t. P(λ¯ w) ¯ = 0,

and in turn E0 ≤ Ehw (λ¯ w) ¯

1 1 n( p − 1) p+1 − λ¯ p+1 w ¯ L p+1 (Rn ) + λ¯ 2 w ¯ 2L 2 (Rn ) = 2( p + 1) p+1 2



n( p − 1) 1 1 p+1 2 2 ¯ − w ¯ L p+1 (Rn ) + w ¯ L 2 (Rn ) ≤λ 2( p + 1) p+1 2

n( p − 1) 1 1 p+1 w ¯ L p+1 (Rn ) + w ≤ − ¯ 2L 2 (Rn ) . 2( p + 1) p+1 2 Notice that in the last inequality we get equality only in the case λ¯ = 1. Moreover by (4.4) and (4.5) we have

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1 n( p − 1) 1 p+1 − w ¯ L p+1 (Rn ) + w ¯ 2L 2 (Rn ) ≤ E0 . 2( p + 1) p+1 2

As a conclusion we deduce that above we have equality everywhere and hence the unique possibility is that λ¯ = 1 and we conclude. Next we show via a Schwartz symmetrization argument that it is not restrictive to assume the minimizing sequence to be radially symmetric. Hence given wk ∈ H 1/2 (Rn ) that satisfies (4.5) (but not necessarily radially symmetric), then we can 1/2 construct another radially symmetric sequence u k ∈ Hrad (Rn ) that satisfies (4.5). We ∗ introduce wk as the Schwartz symmetrization of wk . Notice that we have by standard facts about Schwartz symmetrization that lim inf Ehw (wk∗ ) ≤ E0 and wk∗ 2H 1/2 (Rn ) ≤ k→∞

d( p − 1) ∗ p+1 wk  L p+1 (Rn ) . p+1

(4.7)

Notice that in principle P(wk∗ ) ≤ 0. On the other hands ∃λk ∈ (0, 1]

s.t. P(λk wk∗ ) = 0.

We conclude if we show that λk → 1. In fact in this case it is easy to check that 1/2 λk wk∗ ∈ Hrad (Rn ), λk wk∗ ∈ M and Ehw (λk wk∗ ) − Ehw (wk∗ ) → 0 and hence we conclude by (4.7) that Ehw (λk wk∗ ) → E0 . In order to prove λk → 1 we notice that by (4.4) we have

1 n( p − 1) 1 p+1 p+1 − λk wk∗  p+1 + λ2k wk∗ 22 2( p + 1) p+1 2

1 n( p − 1) 1 p+1 − λ2k wk∗  p+1 + λ2k wk 2L 2 (Rn ) ≤ 2( p + 1) p+1 2

1 n( p − 1) 1 p+1 − λ2k wk  p+1 + λ2k wk 2L 2 (Rn ) = 2( p + 1) p+1 2

E0 ≤ Ehw (λk wk∗ ) =



= λ2k Ehw (wk ) where we used (4.4) at the last step. We deduce that λk → 1 since Ehw (wk ) → E0 .   We can now deduce for every r > 0 the existence of solitary waves belonging to Sr that moreover are minimizers of Ehw constraint to Sr ∩ M. In fact let w be as in Lemma 4.2. Notice that by Lemma 4.1 we get √

−w + w − w p = 0.

Moreover it is clear that w ∈ Ar0 where r0 = w2L 2 (Rn ) . Hence the first part of Theorem 1.3 is proved for r = r0 . The case of a generic r can be achieved by a straightforward rescaling argument.

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4.2 Inflation of H 1/2 -norm for P( f ) < 0 and Ehw ( f ) < I r In this section we follow the approach of [4]. In the sequel the radially symmetric function ϕ : Rn → R is defined as  ϕ(r ) =

r2 2

const

for r ≤ 1; for r ≥ 10.

(4.8)

with ϕ  (t) ≤ 2 for r > 0, and we introduce the rescaled function ϕ R : Rn → R as ϕ R (x) := R 2 ϕ( Rx ). We define the localized virial in the spirit of Ogawa–Tsutsumi [21]  Mϕ (u) = 2 Im

Rn

u∇ϕ ¯ · ∇ud x

(4.9)

In Lemma A.1 of [4] it is shown that Mϕ (u) can be bounded, as follows:   |Mϕ (u)| ≤ C u2H˙ 1/2 (Rn ) + u L 2 (Rn ) u H˙ 1/2 (Rn )

(4.10)

where the constant C depends only on ∇ϕW 1,∞ (Rn ) and on the space dimension. The following Lemma is crucial for our result. 1 (Rn ) we have: Lemma 4.3 (Lemma 2.1, [4]) Let n ≥ 2, for any f ∈ Hrad

d Mϕ (u) = dt



∞

m

1 2



0

2( p − 1) − p+1

R

n

2 2 2 ¯ 4∂k u m (∂lk ϕ)∂l u m − ( ϕ)|u m | d x dm

Rn

(ϕ)|u| p+1 d x

(4.11)

   ˆ ) where u m (t, x) := π1 F −1 ξu(t,ξ and u(t, x) is the unique solution to HW with 2 +m 2 initial condition u(0, x) = f (x). In the sequel we use the following Stein–Weiss inequality for radially symmetric functions due to Rubin [26] in general space dimension n. s (Rn ) we Theorem 4.1 (Rubin [26]) Let n ≥ 2 and 0 < s < n. Then for all u ∈ H˙ rad have: 

1/r r −βr |u(x)| |x| d x ≤ C(n, s, r, β)u H˙ s (Rn ) , (4.12) Rn

where r ≥ 2 and

1 1 −(n − 1) − 2 r 1 1 β −s = + . r 2 n

≤β<

n , r

(4.13) (4.14)

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As a special case of Rubin theorem when the dimension is n ≥ 2, r = p + 1, s = 41 , β = p(−2n+1)+2n+1 we have the following inequality 4( p+1)  Rn

|u(x)|

p+1

|x|

− p(−2n+1)+2n+1 4



1 p+1

dx

≤ Cu H˙ 1/4 (Rn )

(4.15)

2 that holds if 1 < p ≤ 3, which is satisfied since we are assuming 1+ n2 < p < 1+ n−1 .

As a byproduct of (4.15) and by noticing that p(−2n+1)+2n+1 < 0 (this follows by the 4 fact p > 1 + n2 ) we have the following crucial decay: 

|x|≥R

|u(x)|

p+1

dx

≤ CR

p(−2n+1)+2n+1 4

u H˙ 1/4 (Rn )

≤ CR

p(−2n+1)+2n+1 4

u L 22 (Rn ) u H˙21/2 (Rn ) .

p+1 p+1

p+1

(4.16)

We shall also need the following result. Lemma 4.4 For every δ, r > 0 we have sup

{u∈Sr |P (u)<0,Ehw (u)≤Ir −δ}

P(u) < 0.

Proof Assume by the absurd that it is false, then for some δ0 , r0 > 0, we get ∃u k ∈ Sr0 s.t. Ehw (u k ) ≤ Ir0 − δ0 , P(u k ) < 0, P(u k ) → 0. As a first remark we get sup u k  H˙ 1/2 (Rn ) < ∞.

(4.17)

k

In fact it follows by 2 np − n − 2 u k 2H˙ 1/2 (Rn ) = Ehw (u k ) − P(u k ) 2n( p − 1) n( p − 1) and we conclude since lim sup of the r.h.s. is below Ir0 − δ0 . Moreover we have inf u k  H˙ 1/2 (Rn ) > 0. k

(4.18)

In fact notice that by assumption n( p − 1) p+1 u k  L p+1 (Rn ) + P(u k ) p+1 where P(u k ) → 0, P(u k ) < 0.

u k 2H˙ 1/2 (Rn ) =

123

(4.19)

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By combining this fact with the Gagliardo–Nirenberg inequality and by recalling that u k 2L 2 (Rn ) = r0 > 0 we get for suitable universal constants C0 , 0 > 0 (that depend from p, r0 ) 0 u k 2H˙ 1/2 (Rn ) ≤ C0 u k 2+ , H˙ 1/2 (Rn )

and it implies (4.18). Notice also that by a simple continuity argument ∃λk ∈ (0, 1) s.t. P(λk u k ) = 0.

(4.20)

We claim that λk → 1. In fact we get by definition of P we get: λ2k u k 2H˙ 1/2 (Rn ) =

n( p − 1) p+1 p+1 u k  L p+1 (Rn ) . λ p+1 k

(4.21)

By combining the identities (4.21) and (4.19) above we get n( p − 1) p−1 p+1 (λk − 1)u k  L p+1 (Rn ) = P(u k ) → 0. p+1 We conclude that λk → 1 provided that we show inf k u k  L p+1 (Rn ) > 0. Of course it is true otherwise we get 1 u k 2H˙ 1/2 (Rn ) k→∞ 2

0 = lim P(u k ) = lim k→∞

(4.22)

which is in contradiction with (4.18). As a consequence of the fact λk → 1 we deduce λk u k 22 = rk → r0 and hence by (4.20) we get Ehw (λk u k ) ≥ Irk → Ir0 (the last limit follows by elementary considerations). In particular we get Ehw (λk u k ) ≥ Ir0 − δ20 . Moreover Ehw (u k ) − Ehw (λk u k )  1 1  p+1 p+1 1 − λk = (1 − λ2k )u k 2H˙ 1/2 (Rn ) − u k  L p+1 (Rn ) → 0 2 p+1 where we used (4.17) with λk → 1. We get a contradiction since Ehw (λk u k ) ≥ Ir0 − δ20 and Ehw (u k ) ≤ Ir0 − δ0 .   Lemma 4.5 (Localized virial identity for HW) There exists a constant C > 0 such that

p+1 p+1 p(−2n+1)+2n+1 d 4 (4.23) Mϕ R (u) ≤ 4P(u) + C R −1 + R u L 22 (Rn ) u H˙21/2 (Rn ) dt for any u(t, x) radially symmetric solution to HW.

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Proof In [4] it is shown (by choosing s = 21 ) the following estimates: 

∞

4

 m 1/2

0

Rn

∂k u¯ m (∂lk ϕ R )∂l u m d xdm ≤ 2u(t)2H˙ 1/2 (Rn )

and    

∞

 m 1/2

Rn

0

  (2 ϕ R )|u m |2 d xdm  ≤ C R −1 .

p−1) Concerning the last term in (4.11), namely − 2(p+1



Rn (ϕ R )|u|

p+1 d x,

we have

 2( p − 1) (ϕ R )|u| p+1 d x − p + 1 Rn   2( p − 1) 2n( p − 1) p+1 |u| dx − (ϕ R − n)|u| p+1 d x. =− p+1 p + 1 Rn Rn Notice that ϕ R = n on {|x| ≤ R}, hence by recalling (4.16) and summarizing the estimates above we get:  2n( p − 1) d Mϕ R (u) ≤ 2u2H˙ 1/2 (Rn ) − |u| p+1 d x n dt p+1 R

p+1 p+1 p(−2n+1)+2n+1 −1 4 +C R + R u L 22 (Rn ) u H˙21/2 (Rn ) which is equivalent to d Mϕ R (u) ≤ 4P(u) + C dt



R −1 + R

p(−2n+1)+2n+1 4

p+1 p+1 u L 22 (Rn ) u H˙21/2 (Rn ) .  

We can now conclude the proof on the inflation of the norms, in the case that the solution exists globally in time. First notice that by Lemma 4.4 we get P(u(t, x)) < −δ < 0, and we claim that it implies inf u(t, x) H˙ 1/2 (Rn ) > 0

(4.24)

d Mϕ R (u) ≤ 2P(u) − αu2H˙ 1/2 (Rn ) < −αu2H˙ 1/2 (Rn ) . dt

(4.25)

t

and for R sufficiently large

for some constant α > 0. Let us first prove (4.24) and assume by contradiction the existence of a sequence of times tn such that limn→∞ u(tn , x)2H˙ 1/2 = 0. This fact

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implies by Sobolev embedding and conservation of the mass, that P(u(tn , x)) → 0 which contradicts P(u(t, x)) < −δ < 0. Now let us prove (4.25). By using (4.23) it is sufficient to prove that there exists α sufficiently small such that

−1

4P(u) + C

R

Notice that inequality

p+1 2

+R

p(−2n+1)+2n+1 4

u

p+1 2 L 2 (Rn )

u

p+1 2 H˙ 1/2 (Rn )

≤ 2P(u) − αu2H˙ 1/2 (Rn )

< 2 and thanks to (4.24) and conservation of the mass we have the

p(−2n+1)+2n+1

p+1

p+1

4 R −1 + R u L 22 (Rn ) u H˙21/2 (Rn )   p(−2n+1)+2n+1 4 ≤ 4P(u) + C R −1 + R u2H˙ 1/2 (Rn )

4P(u) + C



and hence it suffices to show that   p(−2n+1)+2n+1 4 2P(u) + C R −1 + R u2H˙ 1/2 (Rn ) + αu2H˙ 1/2 (Rn ) < 0

(4.26)

to get (4.25). From the identity Ehw (u) −

r n( p − 1) − 2 2 P(u) = + u2H˙ 1/2 (Rn ) n( p − 1) 2 2n( p − 1)

we get u2H˙ 1/2 (Rn ) ≤

2n( p − 1) 4 Ehw (u) − P(u). n( p − 1) − 2 n( p − 1) − 2

As a consequence we get   p(−2n+1)+2n+1 4 2P(u) + C R −1 + R u2H˙ 1/2 (Rn ) + αu2H˙ 1/2 (Rn )

p(−2n+1)+2n+1 4C 4α 4 R P(u) < 2− − n( p − 1) − 2 n( p − 1) − 2 2nC( p − 1) p(−2n+1)+2n+1 2nα( p − 1) 4 + R Ehw ( f ) + C R −1 Ehw ( f ) + n( p − 1) − 2 n( p − 1) − 2 Notice that we can conclude (4.26) since P(u(t, x)) < −δ and hence it is sufficient to select α very small and R very large. By combining (4.24) with (4.10) we get |Mϕ R (u)| ≤ C(R)u2H˙ 1/2 (Rn ) .

(4.27)

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From (4.25) we deduce that we can select t1 ∈ R such that Mϕ R (u(t)) ≤ 0 for t ≥ t1 and Mϕ R (u(t1 )) = 0. Hence integrating (4.25) we obtain  Mϕ R (u(t)) ≤ −α

t t1

u(s)2H˙ 1/2 (Rn ) ds.

Now by using (4.27) we get the integral inequality    Mϕ (u(t)) ≥ C(R, α) R



t t1

   Mϕ (u(s)) ds, R

which yields an exponential lower bound. 4.3 Instability of A r Given v(x) ∈ Ar we shall show that there exists a sequence λk → 1, λk > 1 such that d/2

d/2

P(λk v(λk x)) < 0, Ehw (λk v(λk x)) < Ehw (v) = Ir . Then by denoting with vk (t, x) the unique solution to HW such that vk (0, x) = d/2 λk v(λk x) we get, by the inflation of norm proved in the previous subsection, that vk (t, x) are unbounded in H 1/2 for large time, despite to the fact that they are arbitrary close to v(x) at the initial time t = 0. Of course it implies the instability of Ar . In order to prove the existence of λk as above, we introduce the functions h : (0, ∞)  λ → P(λd/2 v(λx)) 1 n( p − 1) n( p−1)/2 p+1 λ = λv2H˙ 1/2 (Rn ) − v L p+1 (Rn ) , 2 2( p + 1) g : (0, ∞)  λ → Ehw (λn/2 v(λx)) 1 1 p+1 = λv2H˙ 1/2 (Rn ) + v2L 2 (Rn ) − λn( p−1)/2 v L p+1 (Rn ) . 2 p+1 Notice that since v(x) ∈ M we get h(1) = 0 and hence by elementary analysis of the function h we deduce that h(λ) < 0 for every λ > 1. Moreover again by the fact that v(x) ∈ M we get g  (1) = 0 and since g(1) = Ehw (v) = Ir we deduce that g(λ) < Ir for every λ > 0. It is now easy to conclude the existence of λk with the desired property. Compliance with ethical standards Conflict of interest The authors declare that they have no conflict of interest.

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Appendix In this appendix we prove a global existence result for quartic sNLS with n = 1 and initial condition in H 3/2 (R). It is worth mentioning that the same argument used in [22], where it is treated the one dimensional quartic HW, can be adapted to sNLS. Hence the result stated below can be improved by assuming f (x) ∈ H 1 (R). However we want to give the argument below, since we believe that it is more transparent and slightly simpler. In particular it does not involve the use of fractional Leibnitz rules as in [22]. Moreover, in our opinion, it makes more clear the argument that stands behind the modified energy technique, which is a basic tool in [22] and that hopefully will be a basic tool to deal with other situations (see for instance [23]). Theorem 4.2 Let us fix n = 1 and p = 4. Assume that u(t, x) solves sNLS with initial datum f (x) ∈ H 3/2 (R) and assume moreover that sup

(−T− ( f ),T+ ( f ))

u(t, x) H 1/2 (R) < ∞,

where (−T− ( f ), T+ ( f )) is the maximal interval of existence. Then necessarily T± ( f ) = ∞, namely the solution is global. Proof We have to show that the norm u(t, x) H 3/2 cannot blow-up in finite time. In order to do that we notice that √ i∂t (∂x u) = ( 1 − )∂x u − ∂x (u|u|3 ) ¯ we integrate by parts and we get and hence if we multiply this equation by ∂t (∂x u), the real part, then we obtain: 

√ ( 1 − )∂x u∂t (∂x u)d ¯ x −



∂t (∂x u)∂ ¯ x (u|u|3 )d x     1/4 1/4 =  (1 − ) ∂x u∂t ∂x (1 − ) u¯ d x −  ∂t (∂x u)∂ ¯ x u|u|3 d x R R  − ∂t (∂x u)u∂ ¯ x |u|3 d x

0=

R

R

R

and hence  1 1 d 1/4 2 (1 − ) ∂x u L 2 (R) − ∂t (|∂x u|2 )|u|3 d x 0= 2 dt 2 R  3 −  ∂t (∂x u)u∂ ¯ x |u|2 |u|d x 2 R   1 d 1 1 d 1/4 2 2 3 (1 − ) ∂x u L 2 (R) − |∂x u| |u| d x + |∂x u|2 ∂t (|u|3 )d x = 2 dt 2 dt R 2 R   3 3 −  ∂t (∂x u)u∂ ¯ x uu|u|d ¯ x −  ∂t (∂x u)∂ ¯ x u|u|3 d x. 2 2 R R

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We can continue as follows   1 d 1 1 d |∂x u|2 |u|3 d x + |∂x u|2 ∂t (|u|3 )d x (1 − )1/4 ∂x u2L 2 (R) − 2 dt 2 dt R 2 R   3 3 −  ∂t [(∂x u)] ¯ 2 u 2 |u|d x −  ∂t (|∂x u|2 )|u|3 d x 4 4 R R  d 1 1 1 d 1/4 2 2 3 (1 − ) ∂x u L 2 (R) − |∂x u| |u| d x + |∂x u|2 ∂t (|u|3 )d x = 2 dt 2 dt R 2 R   3 3 d  (∂x u) ¯ 2 u 2 |u|d x +  (∂x u) ¯ 2 ∂t (u 2 |u|)d x − 4 dt 4 R R   3 3 d −  |∂x u|2 |u|3 d x +  |∂x u|2 ∂t (|u|3 )d x. 4 dt 4 R R

0=

Summarizing we get d 5 E(u) = − dt 4



3 |∂x u| ∂t (|u| )d x −  4 R 2



3

R

(∂x u) ¯ 2 ∂t (u 2 |u|)d x.

(4.28)

where 1 5 E(u) = (1 − )1/4 ∂x u2L 2 (R) − 2 4



3 |∂x u| |u| d x −  4 R 2



3

R

(∂x u) ¯ 2 u 2 |u|d x.

Notice that since u(t, x) solves sNLS we deduce that the r.h.s. in (4.28) can be estimated by the following quantity (up to a constant):  R

√ |∂x u|2 | 1 − u||u|2 d x +

 R

|∂x u|2 |u|6 d x

≤ u2W 1,4 (R) u H 1 (R) u2L ∞ (R) + u2H 1 (R) u6L ∞ (R) . Next by using the Gagliardo–Nirenberg inequality u2W 1,4 (R) ≤ Cu H 3/2 (R) u H 1 (R) , and the Brezis–Gallouët inequality (see [6]), together with the assumption on the boundedness of H 1/2 -norm of u(t, x), we can continue the estimate above as follows: r.h.s. (4.28) ≤ Cu H 3/2 (R) u2H 1 (R) ln(2 + u H 3/2 (R) ) + u H 3/2 (R) ln3 (2 + u H 3/2 (R) ) ≤ Cu2H 3/2 (R) ln(2 + u H 3/2 (R) ) + u H 3/2 (R) ln3 (2 + u H 3/2 (R) ).

(4.29)

Notice also that the second and third term involved in the energy E(u) can be estimated as follows: u2H 1 (R) u3L ∞ (R) ≤ Cu H 3/2 (R) ln3/2 (2 + u H 3/2 (R) ).

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By combining the estimate above with (4.29) and (4.28), and by recalling that d 2 dt u L 2 (R) = 0 we deduce u(t)2H 3/2 (R) ≤ Cu(0)2H 3/2 (R) + C sup u(s) H 3/2 (R) ln3/2 (2 + u(s) H 3/2 (R) ) s∈(0,t)



t

+C 0

u(s)2H 3/2 (R) ln(2 + u(s) H 3/2 (R) )

+ u(s) H 3/2 (R) ln3 (2 + u(s) H 3/2 (R) )ds. We conclude by a suitable version of the Gronwall Lemma (see [22] for more details on this point).  

References 1. Agmon, S.: Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2(2), 151–218 (1975) 2. Bellazzini, J., Frank, R.L., Visciglia, N.: Maximizers for Gagliardo-Nirenberg inequalities and related non-local problems. Math. Annalen 360(3–4), 653–673 (2014) 3. Bellazzini, J., Boussaid, N., Jeanjean, L., Visciglia, N.: Existence and stability of standing waves for supercritical NLS with a partial confinement. Commun. Math. Phys. 353(1), 229–251 (2017) 4. Boulenger, T., Himmelsbach, D., Lenzmann, E.: Blowup for fractional NLS. J. Funct. Anal. 271(9), 2569–2603 (2016) 5. Brezis, H., Lieb, H.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88(3), 486–490 (1983) 6. Brezis, H., Gallouët, T.: Nonlinear Schrödinger evolution equations. Nonlinear Anal. Theory Methods Appl. 4, 677–681 (1980) 7. Cazenave, T., Lions, P.L.: Orbital stability of standing waves for some non linear Schrödinger equations. Commun. Math. Phys. 85(4), 549–561 (1982) 8. Cho, Y., Ozawa, T.: Sobolev inequalities with symmetry. Commun. Contemp. Math. 11(3), 355–365 (2009) 9. Cho, Y., Ozawa, T.: On radial solutions of semi-relativistic Hartree equations. Discrete Contin. Dyn. Syst. S 1, 71–82 (2008) 10. Cho, Y., Ozawa, T., Sasaki, H., Shim, Y.: Remarks on the semirelativistic Hartree equations. Discrete Contin. Dyn. Syst. 23(4), 1277–1294 (2009) 11. Christ, M., Kiselev, A.: Maximal functions associated to filtrations. J. Funct. Anal. 179(2), 409–425 (2001) 12. Frank, R.L., Lenzmann, E.: Uniqueness of non-linear ground states for fractional Laplacians in R. Acta Math. 210(2), 261–318 (2013) 13. Frank, R.L., Lenzmann, E., Silvestre, L.: Uniqueness of radial solutions for the fractional Laplacian. Commun. Pure Appl. Math. 69(9), 1671–1726 (2016) 14. Fröhlich, J., Lenzmann, E.: Blowup for nonlinear wave equations describing boson stars. Commun. Pure Appl. Math. 60(11), 1691–1705 (2007) 15. Fröhlich, J., Jonsson, B.L.G., Lenzmann, E.: Boson stars as solitary waves. Commun. Math. Phys. 274(1), 1–30 (2007) 16. Fujiwara, K., Machihara, S., Ozawa, T.: Remark on a semirelativistic equation in the energy space. In: AIMS Proceedings on Dynamical Systems, Differential Equations and Applications, pp. 473–478. Madrid, Spain. https://doi.org/10.3934/proc.2015.0473 17. Ginibre, J., Velo, G.: Smoothing properties and retarded estimates for some dispersive evolution equations. Commun. Math. Phys. 144, 163–188 (1992) 18. Hong, Y., Sire, Y.: On fractional Schrödinger equations in sobolev spaces. Commun. Pure Appl. Anal. 14(6), 2265–2282 (2015) 19. Kato, Tosio: Wave operators and unitary equivalence. Pac. J. Math. 15, 171–180 (1965)

123

740

J. Bellazzini et al.

20. Krieger, J., Lenzmann, E., Raphael, P.: Non dispersive solutions for the L 2 critical half-wave equation. Arch. Ration. Mech. Anal. 209, 61–129 (2013) 21. Ogawa, T., Tsutsumi, Y.: Blow-up of H 1 solution for the nonlinear Schrödinger equation. J. Differ. Equ. 92(2), 317–330 (1991) 22. Ozawa, T., Visciglia, N.: An improvement on the Brezis–Gallouët technique for 2D NLS and 1D half-wave equation. Ann. Inst. Henri Poincare Analyse non lineaire 33, 1069–1079 (2016) 23. Planchon, F., Tzvetkov, N., Visciglia, N.: On the growth of Sobolev norms for NLS on 2d and 3d manifolds. Anal. PDE 10(5), 1123–1147 (2017) 24. Reed, M., Simon, B.: Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic [Harcourt Brace Jovanovich Publishers], New York (1975) 25. Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. Academic [Harcourt Brace Jovanovich Publishers], New York (1975) 26. Rubin, B.S.: One-dimensional representation, inversion and certain properties of Riesz potentials of radial functions. Mat. Zametki 34(4), 521–533 (1983) 27. Secchi, S.: On fractional Schrödinger equations in R N without the Ambrosetti–Rabinowitz condition. Topol. Methods Nonlinear Anal. 47(1), 19–41 (2016) 28. Sickel, W., Skrzypczak, L.: Radial subspaces of Besov and Lizorkin–Triebel classes: extended Strauss lemma and compactness of embeddings. J. Fourier Anal. Appl. 6(6), 639–662 (2000) 29. Skrzypczak, L.: Rotation invariant subspaces of Besov and Triebel–Lizorkin space: compactness of embeddings, smoothness and decay of functions. Rev. Mat. Iberoamericana 18, 267–299 (2002) 30. Strauss, W.: Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55, 149–162 (1977)

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