Existence and stability results on a class of Non Linear Schroedinger Equations in bounded domains with Dirichlet boundary conditions

0.28 MB·English

by  Marco Ghimenti

Existence and stability results on a class of Non
5
1 Linear Schr¨odinger Equations in bounded
0
2 domains with Dirichlet boundary conditions

g
u
A Marco Ghimenti1, Dimitrios Kandilakis2, Manolis Magiropoulos3

9
1
1 Dipartimento di Matematica,

Universit`a di Pisa,
]
P Largo Buonarroti 1/c,
A
56127 Pisa, Italy
h. [e-mail]
t
a
m 2School of Architectural Engineering
[ Techical University of Crete,

73100 Chania, Greece
2
v [e-mail]
9
4
3
5 Technological Educational Institute of Crete
4 Department of Electrical Engineering
0
71500 Heraklion, Crete, Greece
.
1
mageir@staff.teicrete.gr
0
5
1
: Abstract
v
Xi Existence of solution and L2and H1localization results on a class of
Non Linear Schr¨odinger type equations with a bounded nonlinearity are
r
a obtained, for aboundeddomain and with Dirichlet boundaryconditions.
Thekindofstability underdiscussion showsthatthecorrespondingsolu-
tion exhibits features of a solitary wave type.
Keywords: Non Linear Schr¨odinger Equation, stability of solutions,
solitary wave.
2010 Mathematics Subject Classification: 35Q55, 37K45.
The first authoris partial supported by G.N.A.M.P.A.
1
1 Introduction
We study the existence, stability and localization of soliton type solutions for
the Non Linear Schr¨odinger Equation (briefly, NLSE) in the semiclassicallimit
(thatisforh→0+),foraboundeddomainwithDirichletboundaryconditions.
In our framework, the problem takes the form
∂ψ h2 1 ψ
ih =− ∆ψ+ W′(|ψ|) +V(x)ψ , ψ ∈C1(R+,H1(Ω,C))
∂t 2 2hα |ψ| 0 0
ψ(0,x)=φ (x), x∈Ω, (1)
h
ψ(t,x)=0 on R+×∂Ω,
0
Ω ⊂ RN being open and bounded, N ≧ 3, α > 0, where φ (x) ∈ H1(Ω) is
h 0
a suitable initial datum, and V is an external potential. Conditions for the
nonlinear term W and the potential V are to be precised and discussed in the
following sections.
The NLSE in the presence of a potential is largely present in literature. In
particular, it has been extensively studied the effect of the potential V on the
existence and the profile of a stationary solution, that is a solution of the form
ψ(t,x)=U(x)e−hiωt, ω =λ/2, where U solves the equation
1
−h2∆U + W′(U)+2V(x)U =λU.
hα
The first attempt to this direction is the work of Floer and Weinstein [15], for
the one dimensional cubic NLSE (with a generalization for higher dimensions
anddifferentnonlinearityin [21]) where,by meansof aLyapunov- Schmidt re-
duction,itisprovedthat,ifV hasanondegenerateminimum,thenastationary
solutionexists,andthissolutionhasapeaklocatedatthisminimum. Del,Pino
andFelmer[14]showedthatany(possiblydegenerate)minimumofV generates
a stationary solution. We also mention [1, 20], in which similar results are ob-
tained with different techniques.Concerningglobalmethods, in [24] Rabinowitz
proved the existence of a stationary solution with a Mountain Pass argument.
Later,CingolaniandLazzo[13]provedthat the Lusternik - Schnirelmanncate-
goryofthe minimallevelofV givesalowerbound forthe number ofstationary
solutions. The topological approach was also adopted in [2], where a more re-
fined topological invariant is used, and in [9], where the presence of a negative
potential allows the existence of a solution in the so called “zero mass” case.
Another interesting feature is the influence of the domain in the stationary
NLSE, when V = 0. In this case, a single-peaked solution can be constructed.
In [22], Ni and Wei showed that the least energy solution for the equation
−h2∆u+u=up, u>0 in Ω, (2)
with homogeneous Dirichlet boundary condition, has a unique peak, located at
a point P with d(P ,∂Ω) → max d(P,∂Ω). Later,Wei [26] proveda result
h h Ω
h→0+
2
that can be viewed as the converse of the forementioned theorem. Namely, the
authorshowedthatforanylocalmaximumP ofthedistancefromtheboundary
∂Ω, one can constructa single-peakedsolutionof (2) whose peak tends to P as
h→0+. The profileofthesolutionis,uptorescaling,closetothe profileofthe
ground state solution of the limit problem
−∆u+u=up, u>0,
in the whole of RN. We also mention [10] in which the existence of a multi-
peaked solution of (2) is proved.
In the present work we follow a different approach, incorporating and ex-
ploiting ideas found in [4, 6, 7], where the problem (1) had been studied for
the whole of RN for both cases: V = 0 (existence and stability), and V 6= 0
(existence, stability and dynamics).
Wewanttopointouttwomaindifferencesbetweenthispaperand[4]. First,
dealing with a bounded domain and a bounded nonlinearity gives us enough
compactness to easily prove an orbital stability result. In particular our result
is also true for positive nonlinearities, which is forbidden in the whole space by
Pohozaev’s Theorem.
This difference is less evident in what comes after, since, when dealing with
the semiclassical limit, we have to face a limit problem in the whole space RN,
which is the same of [4], so we have to reintroduce some hypothesis.
However, our orbital stability result could be used when studying other
situations, e.g. fixing h and looking for solitons of prescribed, possibly large,
L2 norm (the so called large solitons). If one is interested in these topics, we
recommend the nice paper by Noris, Tavares, Verzini [23], which deals with
orbitalstabilityforsolitonswithprescribedL2 normsinboundeddomains,and
the references therein. We point out that in [23], the authors work with pure
powernonlinearities,sotheycanhavesomecompactnessloss-forL2-criticaland
supercritical powers- even if the domain is bounded. In particular, for critical
and supercritical powers they have orbital stability only for small L2 norms.
The second point in which the bounded domain marks a difference with the
paper [4], is pointed out in the Appendix. In fact, when trying to describe dy-
namics, immediately it appears a repulsive effect of the boundary. The soliton
is affected by a force oriented with the inward normal to the boundary. Unfor-
tunately,the value ofthe forcedepends onthe D1,2 normofthe solutiononthe
boundary, so, at the moment, we were not able to give a quantitative estimate
of this repulsive force, and further efforts are needed.
According to this line of thought, we have divided the present work into
three sections and an appendix:
In Section 2, existence and orbitalstability results are obtained for the case
V =0, by referring to the related eigenvalue problem
−∆U +W′(U) = λU, in Ω
(3)
U ≡ 0, on ∂Ω,
3
given that a solution U(x) in H01(Ω) of (3) results to a solution ψ =U(x)e−iλ2t
of (1), with initial condition ψ(0,x) = U(x). These results are summarized in
Proposition 6. One should notice that the relative proofs work without having
to impose the usual restriction 2 < p < 2+ 4, by relaxing the restriction to
N
2<p<2∗ = 2N instead. It is the boundedness of the domain that allows us
N−2
in doing so.
InSection3,whereweassumethepresenceofanexternalpotential,ourbasic
result,obtainedbymeansofarescalingprocedure,istoproveL2 localizationin
the sense that if we start with an initial datum close to a groundstate solution
U of
1 ω
−h∆U + W′(U) = U, in Ω
hα+1 hα+1
(4)
U ≡ 0, on ∂Ω,
the corresponding solution of (1) will keep its L2 profile along the motion, pro-
vided that h is sufficiently small. Here and in what follows, the restriction
2<p<2+ 4 is imposed, since we need to face a limit problem in RN.
N
In Section 4, an H1 modular localization result is obtained for the case
V 6= 0, and for both cases: the unbounded and the bounded one. When we
work on the whole of RN, we start with a ground state solution U of the RN
1
counterpart of (3), proving that a solution of the RN counterpart of (1), with
initial condition close to U , preserves its basic modular H1 profile as time
1
passes, in the sense that given ε > 0, for all t ≥ 0, the ratio of the squared
L2 norm of |∇u (t,x)| with respect to the complement of a suitable open ball
h
overthe squared L2 norm of |∇u (t,x)| with respect to the whole of RN is less
h
than ε for h sufficiently small, where u (t,x) is taken by the polar expression
h
of ψ(t,x), namely ψ(t,x) = uh(t,x)eish(t,x). The bounded case is treated by
exploiting ideas developed for the L2 problem (Section 3).
Finally, and as we mentioned before, in the Appedix it is described an at-
tempt to study dynamics in the frame of a bounded domain, where we encoun-
tered difficulties due to computational complications related to the action of
∇V on the motion as well as to the repulsive effect of the boundary.
2 The case V = 0
2.1 Existence
For simplicity of the exposition we assume h = 1. As it has been already said,
the case V = 0 is related to problem (3), and a solution u(x) in H1(Ω) of (3)
0
results to a solution ψ =u(x)e−iλ2t of (1) with initial condition ψ(0,x)=u(x).
Notice that a minimizer of
1
J(u)= |∇u|2+W(u) dx
2
ZΩ(cid:18) (cid:19)
4
onS = u∈H1(Ω):kuk =σ ,for somefixedσ >0,isa solutionof(3),
σ 0 L2(Ω)
for suitabnle λ. Thus we focus on theoexistence of such a minimizer. We impose
on W the following conditions:
Condition 1 W is a C1, bounded and even map R→R.
Condition 2 |W′(s)| ≤ c|s|p−1, 2 < p < 2∗ = 2N , where c is a suitable
N−2
positive constant.
Remark 3 Although it is intuitively quite clear the construction of such maps,
an easy concrete example is furnished by choosing W =sin(sp), for p ≥0, and
evenly expanding it on the whole of R. We also stress the fact that a bounded
W is related to the global well posedness results by Cazenave.
Notice that
−∞<µ= inf J(u). (5)
u∈Sσ
If{u }isaminimizing sequenceinS forJ(u),thatisJ(u )→µ,itisevident
n σ n
that {u } is bounded in H1(Ω), thus, up to a subsequence, u ⇀ u ∈ H1(Ω),
n 0 n 0
and u →u in L2(Ω). The latter implies ku k →kuk , thus u∈S .
n n L2(Ω) L2(Ω) σ
Next, we obtain a similar result to Proposition 11 in [4],
Proposition 4 If {w } is a minimizing sequence in S for J, that is J(w )→
n σ n
µ, satisfying the constrained P - S condition, that is, there exists a real sequence
λ of Lagrange multipliers such that
n
−∆w +W′(w )−λ w =σ →0, (6)
n n n n n
then λ is bounded.
n
Proof. Since, as we saw above, w is bounded in H1(Ω), (6) implies
n 0
|∇w |2+W′(w )w −λ w2 dx ≤kσ k kw k →0,
n n n n n n ∗ n H1(Ω)
(cid:12)(cid:12)ZΩ(cid:16) (cid:17) (cid:12)(cid:12) 0
where b(cid:12)(cid:12)y k·k∗ is denoted the dual norm for H(cid:12)(cid:12)01(Ω). We have
|∇w |2+W′(w )w −λ w2 dx =
n n n n n
ZΩ(cid:16) (cid:17)
|∇w |2+2W(w )−2W(w )+W′(w )w −λ w2 dx =
n n n n n n n
ZΩ(cid:16) (cid:17)
2J(w )−λ σ2+ (W′(w )w −2W(w ))dx → 0.
n n n n n
ZΩ
5
Notice that J(w ) is bounded, and because of Condition 2,
n
(W′(w )w −2W(w ))dx ≤
n n n
(cid:12)ZΩ (cid:12)
(cid:12) (cid:12)
|W(cid:12)′(w )w |dx+2 |W(w )|dx(cid:12) ≤
(cid:12) n n n (cid:12)
ZΩ ZΩ
c kw kp +2kmeas(Ω) < +∞,
1 n H1(Ω)
0
where k is an upper bound of |W|. Thus λ is bounded.
n
By Ekeland’s principle, if {u } is a minimizing sequence in S for J(u), we
n σ
mayassumethatitsatisfiestheconstrainedP-Scondition,thatis,thereexists
a real sequence λ so that (6) holds. Because of Proposition 4, λ is bounded,
n n
and the following hold
λ → λ
n
u ⇀ u in H1(Ω)
n 0
u → u in Lp(Ω) for 1≤p<2∗.
n
We have already shown that u∈S . Thus, u6=0. Next, we show that
σ
−∆u+W′(u)=λu. (7)
To this end, if ϕ is a test function, combining the three considerations above
with Condition 2, we have
∇u ∇ϕdx → ∇u∇ϕdx
n
ZΩ ZΩ
W′(u )ϕdx → W′(u)ϕdx
n
ZΩ ZΩ
λ u ϕdx → λ uϕdx,
n n
ZΩ ZΩ
implying (7).
Notice next that due to Condition 2, the Nemytskii operator
W :Lt(Ω)→L1(Ω), 2<t<2∗,
is continuous, whereas u →u in Lt(Ω), for 2<t<2∗. Thus,
n
1
µ≤J(u)= |∇u|2dx+ W(u)dx≤ lim J(u )=µ,
n
2 n→∞
ZΩ ZΩ
proving that J(u)=µ.
This completes the proof for the existence of a non trivial solution of (3),
for suitable λ. In fact, the weak convergence u ⇀ u turns out to be a strong
n
one: Since J(u )−J(u), (W(u )−W(u))dx → 0, we obtain ku k →
n Ω n n H1(Ω)
0
R
6
kuk ,thusprovingthatu →uinH1(Ω). SinceW hasbeenassumedeven,
H1(Ω) n 0
0
we may take a nontrivialnonnegative solutionof (3). By Harnack’sinequality,
thissolutionisstrictlypositiveonΩ. We thusobtainapositivesolutionu∈S
σ
for problem (3), for suitable λ. The wave function
−iωt
ψ(t,x)=u(x)e , ω =λ/2 (8)
is a stationary solution of (1), for h = 1, V ≡ 0, with initial condition φ(x) =
ψ(0,x)=u(x). Evidently, −u(x)e−iωt, ω =λ/2, is a stationary solution of (1),
too.
2.2 Stability
We turn next our attention to the stability of the stationary solution. To this
end, we focus on the reduced form of (1),
∂ψ ψ
2i =−∆ψ+W′(|ψ|) in R+×Ω,
∂t |ψ| 0
ψ(0,x)=φ(x), (9)
ψ(t,x)=0 on R+×∂Ω,
0
by taking, as it was mentioned above, h = 1. The different time slices ψ (x)
t
of each solution of (9), where such a solution may be understood as the time
evolution of some initial condition ψ (x), could be thought of as elements of a
0
proper phase space X ⊂L2(Ω,C), with the set
Γ= u(x)eiθ, θ ∈R/2πZ, u∈S , J(u)=µ= inf J(w) (10)
σ
(cid:26) w∈Sσ (cid:27)
being an invariant (under evolution) manifold of X. Evidently, ±u(x)∈Γ.
To make the description of all this more clear, one should notice that if
ψ (x) isatime sliceofasolutionψ(t,x)of(9), theevolutionmapis definedby
t0
U ψ (x)=ψ (x),
t t0 t0+t
meaning that this time slice might be considered as the initial condition of the
solution ψ (t,x)=ψ(t+t ,x). Now, if u(x)eiθ ∈Γ, then u is a solution of (3),
1 0
with suitable λ, and, at the same time, u(x)eiθ is the initial condition of the
solution ψ(t,x)=u(x)ei(θ−λt/2) of (9). Since u(x)ei(θ−λt/2) ∈Γ for each t>0,
−iωt
the invarianceof Γ follows. We are goingto proveorbital stability of u(x)e ,
ω =λ/2,following the definition oforbitalstability found in[12], meaning that
Γ is stable in the following sense:
∀ε>0, ∃ δ >0 such that if ψ(t,x) is a solution of (9) satisfying
inf kψ(0,x)−wk <δ, then ∀t≧0inf kψ(t,x)−wk <ε. (11)
we∈Γ H01(Ω) we∈Γ H01(Ω)
e e
7
The name orbital could be misleading in the present framework: our non-
linearity does not ensure the uniqueness of the ground state. Moreover, the
problem is not invariant under translations. Thus it could be that the set Γ
reduces to a single function or to a discrete set of ground states (up to multi-
plication by a unitary complex number). Anyhow, we follow the approach by
[12]andby[4],sowewillkeepthe termorbitalstability. InLemmas21and22,
we will see how the orbital stability result leads to a precise description of the
modulus of a solution ψ(t,x) starting from a suitable initial datum.
Notice that Γ is bounded in H1(Ω), since for each of its elements w =
0
w(x)eiθ, w(x) is a constrained minimizer of J, whereas W is bounded. Notice
that we may take w(x)>0.
e
Suppose Γ is not stable. Then ∃ ε > 0, and sequences δ → 0+, ψ (t,x) of
n n
solutions of (9), and t ≥0 such that
n
inf kψ (0,x)−wk <δ , inf kψ (t ,x)−wk ≥ε. (12)
we∈Γ n H01(Ω) n we∈Γ n n H01(Ω)
Notice that the first inequeality of (12) implies e
inf kψ (0,x)−wk <Cδ →0,
n n
we∈Γ L2(Ω)
where C is the Sobolev constant satiesfying k·k ≤ Ck·k . Thus, we
L2(Ω) H1(Ω)
0
may obtain a sequence w in Γ, such that
n
kψ (0,x)−w k →0. (13)
e n n L2(Ω)
We express now ψn(t,x) in polar form,enamely ψn(t,x)=un(t,x)eisn(t,x), with
u (t,x) = |ψ (t,x)|, ∀t ≧ 0. Since kw k = σ, (13) implies that u (0,x) is
n n n n
L2(Ω)
bounded in L2(Ω), and at least up to a subsequence, still denoted by u (0,x),
n
ku (0,x)k → M ≥ 0. Rewritinge(13) in its squared form, and taking into
n
L2(Ω)
consideration that
u (0,x)|w (x)|dx≤ku (0,x)k σ,
n n n
ZΩ L2(Ω)
we take
0≥M2−2Mσ+σ2 =(M −σ)2,
thus obtaining M =σ.
The polar form ψ(t,x)=u(t,x)eis(t,x), turns (9) into the system
∆u W′(u) ∂s 1
− + + + |∇s|2 u=0
2 2 ∂t 2
(cid:18) (cid:19)
∂ u2+∇·(u2∇s)=0, in R+×Ω,
t 0
(14)
u(0,x)eis(0,x) =φ(x), u(t,x)=0 on R+×∂Ω,
0
8
withthetwoequationsof(14)beingtheEuler-Lagrangeequationsoftheaction
functional
1 1 1 ∂s 1
A(u,s)= |∇u|2dxdt+ W(u)dxdt+ + |∇s|2 u2dxdt.
4 2 2 ∂t 2
ZZ ZZ ZZ (cid:18) (cid:19)
(15)
The total energy is given by
1 1
E(ψ)=E(u,s)= |∇u|2+ u2|∇s|2+W(u) dx, (16)
2 2
ZΩ(cid:18) (cid:19)
that is,
1
E(ψ)=E(u,s)=J(u)+ u2|∇s|2dx (17)
2
ZΩ
Independenceoftimefortheenergyandforthechargeimplythatforasolution
ψ(t,x)=u(t,x)eis(t,x) of (9), it holds
d
u(t,x)2dx=0 (18)
dt
ZΩ
d
E(u,s)=0. (19)
dt
Equivalently, (18) and (19) can be expressed as
kψ(t,x)k =kφ(x)k (20)
L2(Ω) L2(Ω)
E(ψ(t,x))=E(φ(x)) (21)
for all t≥0. Noteworthy, for stationary solution, (17) yields E(ψ)=J(u).
Returning to the sequence ψ (t,x) satisfying (12), we may assume, as we
n
saw, that ku (0,x)k →σ, that is ku (t,x)k →σ, for t≥0, because of
n n
L2(Ω) L2(Ω)
(20). We want to show that {u } is a minimizing sequence for the functional
n n
J on the constraint kuk =σ.
L2(Ω)
Oneshouldnoticethatthefirstinequalityof(12),combinedwiththebound-
edness of Γ ensure that ψ (0,x) is bounded in H1(Ω). Since W is bounded,
n 0
(17) ensures that E(ψ (0,x)) is bounded, and because of (21), E(ψ (t,x)) is
n n
bounded, for all n and all t ≥ 0. In particular, E(ψ (t ,x)) is bounded. A
n n
new application of (17), ensures now that u (t ,x) is bounded in H1(Ω). The
n n 0
sequence u (t ,x) = α u (t ,x), where α = σ , is in S . We
n n n n n n kun(tn,x)kL2(Ω) σ
have, writing for simplicity u , u instead of u (t ,x), u (t ,x), respectively,
n n n n n n
b
b b
9
for suitable l =l (x)∈(0, 1), and because of Condition 2,
n n
1
|J(u )−J(u )| ≤ |α2 −1| |∇u |2dx+ |W(u )−W(u )|dx
n n 2 n n n n
ZΩ ZΩ
1
b = |α2 −1| |∇u |2dx b
2 n n
ZΩ
+|α −1| |u W′(l u +(1−l )u )|dx
n n n n n n
ZΩ
1
≤ |α2 −1| |∇u |2dx b
2 n n
ZΩ
+|α −1| c[l +(1−l )α ]p−1|u |pdx
n n n n n
(cid:26)ZΩ (cid:27)
→ 0,
sinceintherighthandsideofthelastinequality,thetwosummandsareproducts
of a zero sequence by a bounded one. Thus, J(u )−J(u ) → 0. We return
n n
now to
kψ (0,x)−w k →0, (22)
n n b
H01(Ω)
which, as a result of the triangle inequality combined with the boundedness of
e
kψ (0,x)k +kw (x)k , readily gives
n n
H01(Ω) H01(Ω)
kψ (0,x)k2 −kw (x)k2 →0,
n n
H01(Ω) H01(Ω)
that is,
|∇u (0,x)|2+u2(0,x)|∇s (0,x)|2−|∇w (x)|2 dx→0. (23)
n n n n
ZΩh i
We claim that
u2(0,x)|∇s (0,x)|2dx→0. (24)
n n
ZΩ
If not so, up to a subsequence,
|∇u (0,x)|2−|∇w (x)|2 dx→k <0. (25)
n n
ZΩh i
Combining L1 convergenceof u (0,x)−w (x) to 0, with Condition 2, we have
n n
[W(u (0,x))−W (w (x))]→0. (26)
n n
ZΩ
Now (25) and (26) give
J(u (0,x))−J(w (x))→k/2<0. (27)
n n
10