On the stability of ground states in 4D and 5D nonlinear Schrodinger equation including subcritical cases

Abstract

We consider a class of nonlinear Schrodinger equation in four and five space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in L2) nonlinearities. We show that the center manifold formed by localized in space periodic in time solutions (bound states) is an attractor for all solutions with a small initial data. The proof hinges on dispersive estimates that we obtain for the time dependent, Hamiltonian, linearized dynamics around a one parameter family of bound states that "shadows" the nonlinear evolution of the system. The methods we employ are an extension to higher dimensions, hence different linear dispersive behavior, and to rougher nonlinearities of our previous results [10, 11, 7]. Comment: 40 pages

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arXiv:0906.3732v1 [math.AP] 19 Jun 2009
On the stability of ground states in 4D and 5D nonlinear
Schr¨ odinger equation including subcritical cases
E. Kirr and¨O. Mızrak∗
June 19, 2009
Abstract
We consider a class of nonlinear Schr¨ odinger equation in four and five space dimensions with
an attractive potential. The nonlinearity is local but rather general encompassing for the first time
both subcritical and supercritical (in L2) nonlinearities. We show that the center manifold formed
by localized in space periodic in time solutions (bound states) is an attractor for all solutions
with a small initial data. The proof hinges on dispersive estimates that we obtain for the time
dependent, Hamiltonian, linearized dynamics around a one parameter family of bound states that
“shadows” the nonlinear evolution of the system. The methods we employ are an extension to
higher dimensions, hence different linear dispersive behavior, and to rougher nonlinearities of our
previous results [10, 11, 7].
1 Introduction
In this paper we study the long time behavior of solutions of the nonlinear Schr¨ odinger equation (NLS)
with potential in four and five space dimensions (4-d and 5-d):
i∂tu(t,x) = [−∆x+ V (x)]u + g(u),
u(0,x) = u0(x)
t ∈ R,x ∈ RN,N = 4,5 (1.1)
(1.2)
where the local nonlinearity is constructed from the real valued, odd, C1function g : R ?→ R which is
twice differentiable except maybe at zero and satisfies:
g(0) = 0, g′(0) = 0,|g′′(s)| ≤ C(|s|α1−1+ |s|α2−1), for s ?= 0, (1.3)
with
α0(N) =2 − N +√N2+ 12N + 4
2N
< α1≤ α2<
4
N − 2
for N=4, 5.(1.4)
g is then extended to a complex function via the gauge symmetry:
g(eiθz) = eiθg(z),g(¯ z) = g(z). (1.5)
Note that g is not necessarily twice differentiable at 0, e.g. g(z) = |z|
We are going to show that the manifold of periodic solutions of (1.1) (center manifold) is a global
attractor for all small initial data. More precisely, for u0∈ H1∩ L
the solution of (1.1)-(1.2) can be decomposed as follows:
5
6z.
α2+2
α2+1with sufficiently small norm
u(t,x) = eiθ(t)ψE(t)(x) + r(t,x)
∗Department of Mathematics, University of Illinois at Urbana-Champaign
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where for each fixed time t1∈ R, E = E(t1), we have that uE(t,x) = e−iEtψE(x) is a periodic solution
of (1.1) and, as |t| → ∞, r(t) ∈ H1(RN) converges strongly to zero in Lp(RN), 2 < p < 2N/(N − 2),
spaces and weakly in H1(RN), see Section 3 for more details. We can also show that the full dynamics
converges to a certain periodic solution, i.e. ψE(t)→ ψE±∞, for t → ±∞, provided we restrict the range
of nonlinearities to the supercritical regime α1> 4/N, see Corollary 3.2. In this case we generalize the
results in [19, 15].
Moreover, for the remaining range α0(N) < α1? 4/N, we could show a type of asymptotic stability
for periodic solutions (bound states) of (1.1). In other words, if e−iEtψE(x), ψE?≡ 0 is a (small) periodic
solution of (1.1) then there exists ε > 0 depending on ψEsuch that for all initial data u0∈ H1∩L
satisfying
inf
α2+2
α2+1
0≤θ<2π?u0− eiθψE?
H1∩L
α2+2
α2+1< ε(ψE)
the solution of (1.1)-(1.2) converges to a periodic solution (close to e−iEtψE(x)) strongly in Lp,2 <
p < 2N/(N −2), spaces and weakly in H1, as t → ±∞. The proof of this result is left for another paper
[6] because it involves a different decomposition of the dynamics and a more delicate way to obtain
the linear estimates similar to the ones in Section 4. It has the advantage that it can be generalized
to large periodic solutions and the disadvantage that it only describes the evolution of initial data in a
conic like neighborhood (since ε depends on ψE) of the set of periodic solutions (center manifold) with
the zero solution removed. In fact the choice of ε is such that the solution stays away from zero, the
point where the center manifold fails to be C2smooth. Since the dynamics only sees the C2smooth
part of the center manifold a better decomposition of the dynamics can be employed, see [7, 11], and
convergence to a periodic solution follows.
The main contribution of our paper is to describe the long time evolution of all small initial data for
rather general nonlinearities including for the first time the subcritical ones α1< 4/N, see Section 3.
We accomplish this by using a time-dependent projection of the solution of (1.1)-(1.2) onto the center
manifold of periodic solutions described in Section 2 and (3.3). We first prove dispersive estimates
for the propagator of the linearized equation at the time-dependent projection, see Section 4. We
then estimate the error between the actual solution and its projection onto the center manifold via a
Duhamel principle with respect to the linearized operator at the projection and a fixed point argument
for the resulting integral equation, see (3.8). Since the operator on its right hand side contains no
linear terms in the error, we are able to show that it is contractive in appropriately chosen Banach
spaces for a large spectrum of nonlinearities g. This is in contrast with the approach in [15, 19] where
linear and nonlinear terms had to be estimated at the same time, the linear ones requiring the use
of L2weighted (localized) estimates which applied to the non-localized, nonlinear terms forced the
assumption α1> 4/N. In the current approach, as in our previous 3D and 2D results, see [7, 10, 11],
we completely separate estimates for nonlinear terms from the ones for linear terms and use methods
tailored for each of them.
The most difficult part is to obtain dispersive estimates for the propagator of the time-dependent
linearized operator at the projections onto the center manifold, see Section 4. While estimates for the
Schr¨ odinger group of operators:
?e−i(−∆+V (x))tPc?Lp′?→Lp ? Cp|t|−N(1
are well known, see [4] and references therein, they are almost non-existent when the potential V
depends on time V = V (t,x). This is to be expected since the time-dependence of V is the quantum
mechanical analog of the parametric forcing in ordinary differential equations and, in principle, can
lead to very different behavior compared to the time-independent case, see [20, 8, 9, 12, 2, 3]. However,
in the absence of resonant phenomena one might expect similarities between the two dynamics. Indeed,
this is the case in [16] which cannot be generalized to our situation mostly because of the complex-
valued potential, see (3.7) and (2.4). To overcome this issues we use smallness and localization of the
time dependent terms (4.6)-(4.7) to first obtain dispersive estimates in weighted (localized) norms, see
2−1
p),1/p′+ 1/p = 1, 2 ? p ? ∞ (1.6)
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Theorem 4.1. Then in Theorem 4.2 we rely on the integrability in time of the group of operators
generated by the nearby time independent operator −i(−∆ + V (x)), see (1.6) with p > 2N/(N − 2)
and t ? 1, to remove the weights and obtain dispersive estimates in non-localized Lpnorms. But the
integrability in time for t ? 1 comes at the cost of a non-integrable singularity at t = 0 which we remove
by using cancelations in highly oscillatory integrals. The method is similar to the one we employed in
[7], see also [10, 11] for an alternate way of dealing with the singularity.
In a nutshell the results in this paper rely on shadowing the actual solution of (1.1)-(1.2) via a curve
on the central manifold of periodic solutions for (1.1). Essential in showing that the distance between
the solution and its shadow goes to zero are the new, apriori, dispersive estimates for the propagator of
the linearized equation along the shadowing curve. In this regard the paper is an extension to higher
dimensions, hence different linear dispersive behavior, and to rougher nonlinearities of our previous
results in two respectively three space dimensions [10, 11, 7].
Notations: H = −∆ + V ;
Lp= {f : RN?→ C | f measurable and
the standard norm in these spaces;
< x >= (1 + |x|2)1/2, and for σ ∈ R, L2
of functions f(x) such that < x >σf(x) are square integrable endowed with the norm ?f(x)?L2
x >σf(x)?2;
?f,g? =?
Pcis the projection on the continuous spectrum of H in L2;
ˆ u denotes the Fourier transform of the temperate distribution u;
Hs, s ∈ R denote the Sobolev spaces of temperate distributions u such that (1 + |ξ|2)s/2ˆ u(ξ) ∈
L2(RN) with norm ?u?Hs = ?(1+|ξ|2)s/2ˆ u(ξ)?L2. Note that for s = n, n a natural number, this spaces
coincide with the space of measurable functions having all distributional partial derivatives up to order
n in L2.
?
RN|f(x)|pdx < ∞}, ?f?p=??
σdenotes the L2space with weight < x >2σ, i.e. the space
RN|f(x)|pdx?1/pdenotes
σ= ? <
RNf(x)g(x)dx is the scalar product in L2where z = the complex conjugate of the complex
number z;
2 The Center Manifold
The center manifold is formed by the collection of periodic solutions for (1.1):
uE(t,x) = e−iEtψE(x) (2.1)
where E ∈ R and 0 ?≡ ψE∈ H2(RN) satisfy the time independent equation:
[−∆ + V ]ψE+ g(ψE) = EψE
Clearly the function constantly equal to zero is a solution of (2.2) but (iii) in the following hypotheses
on the potential V allows for a bifurcation with a nontrivial, one parameter family of solutions:
(2.2)
(H1) Assume that
(i) V (x) sutisfies the following properties:
1. < x >ρV (x) : Hη→ Hη, for some ρ > N + 4 and η > 0;
2. ∇V ∈ Lp(RN) for some 2 ? p ? ∞ and |∇V (x)| → 0 as |x| → ∞;
3. the Fourier transform of V is in L1.
(ii) 0 is a regular point1of the spectrum of the linear operator H = −∆ + V acting on L2.
1see [18, Definition 7] or Mµ= {0} in relation (3.1) in [13]
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(iii) H acting on L2has exactly one negative eigenvalue E0 < 0 with corresponding normalized
eigenvector ψ0. It is well known that ψ0(x) is exponentially decaying as |x| → ∞, and can be
chosen strictly positive.
Conditions (i) and (ii) guarantee the applicability of dispersive estimates in [13] and [4] to the Schr¨ odinger
group e−iHtPc. Condition (i)2. implies certain regularity of the nonlinear bound states while (i)3. al-
lows us to use commutator type inequalities, see (4.14) and [7, Theorem 5.2]. All these are needed
to obtain estimates for the semigroup of operators generated by our time dependent linearization, see
Theorems 4.1 and 4.2 in section 4. In particular (i)1. implies the local well posedness in H1of the
initial value problem (1.1)-(1.2), see section 3.
By the standard bifurcation argument in Banach spaces [14] for (2.2) at E = E0, condition (iii)
guaranteesexistence of nontrivial solutions. Moreover, these solutions can be organized as a C1manifold
(center manifold):
Proposition 2.1 There exist δ > 0, the C1function
h : {a ∈ C : |a| ≤ δ} ?→ H2∩ L2
σ,σ ∈ R
and the C1function E : [−δ,δ] ?→ R such that for |E − E0| ≤ δ and |?ψ0,ψE?| ≤ δ, ?ψE−
?ψ0,ψE?ψ0?H2 ≤ δ the eigenvalue problem (2.2) has a unique non-zero solution up to multiplication
with eiθ, θ ∈ [0,2π), which can be represented as a center manifold:
ψE= aψ0+ h(a), E = E(|a|),
See [10, section 2] for details.
?ψ0,h(a)? = 0,h(eiθa) = eiθh(a), for |a| ≤ δ. (2.3)
Remark 2.1 By regularity methods, see for example [1, Theorem 8.1.1], one can show ψE ∈ H3∩
W2,p, 2 ? p < ∞. Hence by Sobolev imbeddings both ψE and ∇ψE are continuous and converge to
zero as |x| → ∞. Moreover comparison techniques for elliptic equations, see [7, Section 5.2], imply
exponential decay, i.e. for each 0 < A < −E there exists a constant CA> 0 such that:
√A|x|ψE(x)?L∞ ≤ CA?ψE(x)?L∞ < ∞, and
Remark 2.2 By variational methods, see for example [17], one can show that the real valued solutions
of (2.2) do not change sign. Then Harnack inequality for H2?C(RN) solutions of (2.2) implies that
positive or strictly negative.
?e
?e
√A|x|∇ψE(x)?L∞ ≤ CA?∇ψE(x)?L∞ < ∞.
these real solution cannot take the zero value. Hence ψE given by (2.3) for a ∈ R is either strictly
In section 4 we also need some smoothness for the effective (linear) potential induced by the non-
linearity which for the real valued bound states is:
Dg|ψE[u + iv] = g′(ψE)u + ig(ψE)
ψE
v,ψE> 0(2.4)
while for an arbitrary bound state˜ψE= eiθψE, ψE> 0 we have via the rotational symmetry of g, see
(1.5),
Dg|˜ψE[β] = eiθDg|ψE[e−iθβ].
(H2) Assume that for the positive solution of (2.2) we have?
for the Fourier transform of the function f.
(2.5)
g′(ψE),?
g(ψE)
ψE
∈ L1(RN) whereˆf stands
In concrete cases the hypothesis may be checked directly using the regularity of ψE, the solution of an
uniform elliptic e-value problem. In general we can prove the following result:
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Proposition 2.2 If the following holds:
(H2’) g restricted to reals has four derivatives except at zero and
?
Then for the positive solution of (2.2), ψE, we have?
|g(m)(s)| < C
1
sm−1−α1+
1
sm−1−α2
?
, for m = 3,4, and s > 0.
g′(ψE) ∈ L1and?
g(ψE)
ψE
∈ L1.
Proof:
??
g′(ψE)?L1 =
???
1
(1 + |ξ|2)
3
2(1 + |ξ|2)
3
2 ?
g′(ψE)
???
L1≤
???
1
(1 + |ξ|2)
?
3
2
? ??
∈L2(RN)
???
L2?(1 + |ξ|2)
?
3
2 ?
g′(ψE)
???
∈L2⇔g′(ψE)∈H3
?L2
So it suffices to show that g′(ψE) ∈ H3and
∇g′(ψE) = g′′(ψE)∇ψE
∆g′(ψE) = g′′′(ψE)|∇ψE|2+ g′′(ψE)∆ψE
= g′′′(ψE)|∇ψE|2+ g′′(ψE)[(V − E)ψE+ g(ψE)]
∇3g′(ψE) = g(4)(ψE)|∇ψE|2∇ψE+ 3g′′′(ψE)∇ψE∆ψE+ g′′(ψE)∇3ψE
= g(4)(ψE)|∇ψE|2∇ψE+ 3g′′′(ψE)∇ψE[(V − E)ψE+ g(ψE)] + g′′(ψE)∇[(V − E)ψE+ g(ψE)]
and, using (1.3),
g(ψE)
ψE
∈ H3. We have:
|g′(ψE)| ≤ C(|ψE|α1+ |ψE|α2)
|∇g′(ψE)| ≤ C|∇ψE|(
1
|ψE|1−α1+
1
|ψE|2−α1+
1
|ψE|3−α1+
1
|ψE|1−α2)
1
|ψE|2−α2) + C(|ψE|α1+ |ψE|α2)(|V | + |E|) + C(|ψE|2α1+ |ψE|2α2)
1
|ψE|3−α2) + 2C|∇ψE|(|V | + |E|)(
+ C|∇V |(|ψE|α1+ |ψE|α2) + 2C|∇ψE|(|ψE|2α1−1+ |ψE|2α2−1)
g(ψE)
ψE
as follows:
|∆g′(ψE)| ≤ C|∇ψE|2(
|∇3g′(ψE)| ≤ C|∇ψE|3(
1
|ψE|1−α1+
1
|ψE|1−α2)
Similarly we get the estimates for
|g(ψE)
ψE
|∇g(ψE)
ψE
|∆g(ψE)
ψE
|∇3g(ψE)
| ≤ C(|ψE|α1+ |ψE|α2)
| ≤ C|∇ψE|(
1
|ψE|1−α1+
1
|ψE|2−α1+
1
|ψE|3−α1+
1
|ψE|1−α2)
1
|ψE|2−α2) + C(|ψE|α1+ |ψE|α2)(|V | + |E|) + C(|ψE|2α1+ |ψE|2α2)
1
|ψE|3−α2) + 2C|∇ψE|(|V | + |E|)(
+ C|∇V |(|ψE|α1+ |ψE|α2) + 2C|∇ψE|(|ψE|2α1−1+ |ψE|2α2−1)
Now, we will use the following bounds for ψEand ∇ψE, see [7, Section 5.2]. For any 0 < A < −E <
A2and any 0 < A1< −E there exist the constants CA, CA1, CA2> 0 such that:
CA2e−√A2|x|≤ ψE(x) ≤ CAe−√A|x|,
| ≤ C|∇ψE|2(
ψE
| ≤ C|∇ψE|3(
1
|ψE|1−α1+
1
|ψE|1−α2)
for all x ∈ RN,
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