Existence and stability of standing waves for nonlinear fractional Schr\"odinger equation with logarithmic nonlinearity

0.22 MB·

by  Alex Hernandez Ardila

EXISTENCE AND STABILITY OF STANDING WAVES FOR
NONLINEAR FRACTIONAL SCHRO¨DINGER EQUATION WITH
LOGARITHMIC NONLINEARITY
7
1
0
2 Alex H. Ardila

n DepartmentofMathematics, IME-USP,UniversidadedeS˜aoPaulo
a CidadeUniversit´aria,CEP05508-090, S˜aoPaulo,SP,Brazil
J

0 Abstract. In this paper we consider the nonlinear fractional logarithmic
2
Schro¨dingerequation. Byusingacompactnessmethod,weconstructaunique

globalsolutionoftheassociatedCauchyprobleminasuitablefunctionalframe-
]
P work. Wealsoprovetheexistenceofgroundstatesasminimizersoftheaction
ontheNeharimanifold. Finally,weprovethatthesetofminimizersisastable
A
set for the initial value problem, that is,a solution whoseinitial data is near
. thesetwillremainnearitforalltime.
h
t
a 1. Introduction
m
This paperis concernedwiththe fractionalnonlinearSchr¨odingerequationwith
[
logarithmic nonlinearity

3
i∂ u ( ∆)su+ uLog u2 =0, (1.1)
v t
− − | |
3 where 0 < s < 1 and u = u(x,t) is a complex-valued function of (x,t) RN R,
6 ∈ ×
N 2. The fractional Laplacian ( ∆)s is defined via Fourier transform as
3 ≥ −
1 [( ∆)su](ξ)= ξ 2s u(ξ), (1.2)
0 F − | | F
. where the Fourier transform is given by
1
1
0 u(ξ)= u(x)e−iξ·xdx. (1.3)
7 F (2π)N/2 ZRN
1
ThefractionalLaplacian( ∆)s isaself-adjointoperatoronL2(RN)withquadratic
:
v formdomainHs(RN)and−operatordomainH2s(RN). Moreover,thefollowingspec-
i
X tralpropertiesof( ∆)s areknown: σess(( ∆)s)=[0, )andσp(( ∆)s)= (see,
− − ∞ − ∅
e.g., [26, Example 3.3]). The nonlocal operator ( ∆)s can be seen as the infini-
r
a tesimal generatorsof L´evystable diffusion processe−s(see [2]). Fractionalpowers of
theLaplacianariseinanumerousvarietyofequationsinmathematicalphysicsand
relatedfields; see,e.g., [2, 3, 22] andreferencestherein. Recently, a greatattention
hasbeenfocusedonthestudyofproblemsinvolvingthefractionalLaplacianfroma
puremathematicalpointofview. Concerningthe fractionalSchr¨odingerequations,
let us mention [15, 10, 19, 21, 20, 18, 5, 16, 23, 24, 28].
Thepresentpaperisdevotedtotheanalysisofexistenceandstabilityofstanding
waves of NLS (1.1). If the fractional Laplacian in (1.1) is replaced by a standard
Laplacian,this problem is well-knownand described in detail in [4, 6, 11, 14, 7, 8].
In this case, one can show that that there exists a unique (up to translations and
2010 Mathematics Subject Classification. 35Q55;35Q40.
Keywordsandphrases. FractionallogarithmicSchro¨dingerequation;standingwaves;stability.
Theauthor wassupportedbyCNPq-Brazil.
1
2 ALEXH.ARDILA
phase shifts) ground state and it is orbitally stable. The classical logarithmic NLS
equation was proposed by Bialynicki-Birula and Mycielski [6] in 1976 as a model
of nonlinear wave mechanics and has important applications in quantum mechan-
ics, quantum optics, nuclear physics, open quantum systems and Bose-Einstein
condensation (see e.g. [25, 29] and the references therein).
The energy functional E associated with problem (1.1) is
1 1
E(u)= ( ∆)s/2u2dx u2Log u2dx. (1.4)
2ZRN | − | − 2ZRN | | | |
Unfortunately, due to the singularity of the logarithm at the origin, the functional
fails to be finite as well of class C1 on Hs(RN). Due to this loss of smoothness,
it is convenient to work in a suitable Banach space endowed with a Luxemburg
type norm in order to make functional E well defined and C1 smooth. This space
allows to control the singularity of the logarithmic nonlinearity at infinity and at
the origin. Indeed, we consider the reflexive Banach space (see Section 2)
Ws(RN)= u Hs(RN): u2Log u2 L1(RN) , (1.5)
∈ | | | | ∈
n o
thentheenergyfunctionalE iswell-definedandofclassC1 onWs(RN). Moreover,
fromLemma 2.4,we havethat the operatoru ( ∆)su uLog u2 is continuous
fromWs(RN)to W−s(RN). Here, W−s(RN) i→s th−e duals−paceof|W|s(RN). There-
fore, if u C(R,Ws(RN)) C1(R,W−s(RN)), then equation (1.1) makes sense in
W−s(RN)∈. ∩
The nextpropositiongivesaresultonthe existenceofweaksolutionsto (1.1)in
the energy space Ws(RN). The proof is contained in Section 3.
Proposition1.1. Foranyu Ws(RN),thereisauniquesolutionu C(R,Ws(RN))
0
C1(R,W−s(RN)) of (1.1)such∈that u(0)=u and sup u(t) ∈ < . Fur- ∩
0 t∈Rk kWs(RN) ∞
thermore, the conservation of energy and charge holds; that is,
E(u(t))=E(u ) and u(t) 2 = u 2 for all t R.
0 k kL2 k 0kL2 ∈
In this paper we study the existence and stability of standing wavessolutions of
(1.1) of the form u(x,t) = eiωtϕ(x), where ω R and ϕ Ws(RN) is a complex
∈ ∈
valued function which has to solve the following stationary problem
( ∆)sϕ+ωϕ ϕLog ϕ2 =0, x RN. (1.6)
− − | | ∈
For ω R, we define the following functionals of class C1 on Ws(RN):
∈
1 ω+1 1
S (u)= ( ∆)s/2u2dx+ u2dx u2Log u2dx,
ω
2ZRN | − | 2 ZRN | | − 2ZRN | | | |
I (u)= ( ∆)s/2u2dx+ω u2dx u2Log u2dx.
ω
ZRN | − | ZRN | | −ZRN | | | |
Note that (1.6) is equivalent to S′(ϕ)=0, and I (u)= S′(u),u is the so-called
ω ω h ω i
Nehari functional. It was shown in [18] that the problem (1.6) admits a sequence
of weak solutions u Hs(RN) with S (u ) + as n + .
n ω n
∈ → ∞ → ∞
From the physical point viewpoint, an important role is played by the ground
state solution of (1.6). We recall that a solution ϕ Ws(RN) of (1.6) is termed
∈
as a groundstate if it has some minimal action among allsolutions of (1.6). To be
FRACTIONAL LOGARITHMIC SCHRO¨DINGER EQUATION 3
more specific, we consider the minimization problem
d(ω)=inf S (u): u Ws(RN) 0 ,I (u)=0
ω ω
∈ \{ }
1 (1.7)
= in(cid:8)f u 2 :u Ws(RN) 0 ,I (u(cid:9))=0 ,
2 k kL2(RN) ∈ \{ } ω
n o
and define the set of ground states by
= ϕ Ws(RN) 0 :S (ϕ)=d(ω), I (ϕ)=0 .
ω ω ω
G ∈ \{ }
The set u Ws(R(cid:8)N) 0 ,I (u)=0 is calledthe Neharimanifo(cid:9)ld. Notice that
ω
∈ \{ }
the above set contains all stationary points of S .
ω
(cid:8) (cid:9)
The existence of minimizers for (1.7) will be obtained as a consequence of the
stronger statement that any minimizing sequence for (1.7) is, up to translation,
precompact in Ws(RN). We will show the following theorem in Section 4.
Theorem 1.2. Let N 2, ω R and 0 < s < 1. Let u Ws(RN) be a
n
minimizing sequence for≥d(ω). ∈Then there exists a family{(y }) ⊆ RN such that
n
u ( y ) contents a convergent subsequence in Ws(RN). In⊂particular, this
n n
i{mpli·e−s that} is not empty set for any ω R.
ω
G ∈
We remark that for any u , there exists a Lagrange multiplier Λ R such
ω
∈ G ∈
that S′(u) = ΛI′(u). Thus, we have S′(u),u = Λ I′(u),u . The fact that
ω ω h ω i h ω i
S′(u),u = I (u) = 0 and I′(u),u = 2 u 2 < 0, implies Λ = 0; that is,
h ω i ω h ω i − k kL2(RN)
S′(u)=0. Therefore, u satisfies (1.6).
ω
Now we are ready to state our main result, which is a direct consequence of the
result of relative compactness.
Theorem 1.3. Let N 2, ω R and 0 < s < 1. Then the set G is Ws(RN)-
ω
≥ ∈
stable with respect to NLS (1.1); that is, for arbitrary ǫ>0, there exist δ >0 such
that for any u Ws(RN), if
0
∈
ψi∈nGfωku0−ψkWs(RN) <δ,
thenthesolutionu(x,t)oftheCauchyproblem (1.1)withtheinitialdatau satisfies
0
ψi∈nGfωku(·,t)−ψkWs(RN) <ǫ, for all t≥0.
The rest of the paper is organized as follows. In Section 2, we analyse the
structure of the energy space Ws(RN). Moreover, we recall several known results,
whichwillbeneededlater. InSection3,wegiveanideaoftheproofofProposition
1.1. In Section 4 we prove, by variational techniques, the existence of a minimizer
for d(ω). The stability result is proved in Section 5. In the Appendix we list some
properties of the Orlicz space associated with Ws(RN).
Notation. The space L2(RN,C) will be denoted by L2(RN) and its norm by
L2. , isthedualitypairingbetweenX′andX,whereX isaBanachspaceand
k·k h· ·i
X′ is its dual. Finally, 2∗ := 2N/(N 2s) for N 2 and 0 < s < 1. Throughout
s − ≥
thispaper,theletterCwilldenotepositiveconstantswhosevaluemaychangefrom
line to line.
4 ALEXH.ARDILA
2. Preliminaries and functional setting
For the sake of self-containedness, we first provide some basic properties of the
fractional Sobolev spaces Hs(RN), which will be needed later. Consider the frac-
tional order Sobolev space
Hs(RN)= u L2(RN): (1+ ξ 2s)uˆ(ξ)2dξ < ,
(cid:26) ∈ ZRN | | | | ∞(cid:27)
whereuˆ= (u). Thenormisdefinedby u 2 = (uˆ(ξ)2+ ξ 2s uˆ(ξ)2)dξ.
F k kHs(RN) RN | | | | | |
Notice that, by Plancherel’s theorem we have
R
u 2 = uˆ(ξ)2dξ+ ξ 2s uˆ(ξ)2dξ
k kHs(RN) ZRN | | ZRN | | | |
\
= u(x)2dx+ ( ∆)2su(ξ)2dξ
ZRN | | ZRN | − |
= u(x)2dx+ ( ∆)2su(x)2dx< ,
ZRN | | ZRN | − | ∞
for every u Hs(RN). Moreover, the space Hs(RN) is continuously embedded
into Lq(RN)∈for any q [2,2∗] and compactly embedded into Lq (RN) for any
∈ s loc
q [2,2∗), where 2∗ =2N/N 2s. See [27] for more details.
∈Let 0s< s < 1 asnd Ω a sm−ooth bounded domain of RN, we define Hs(Ω) as
follows
u(x) u(y)
Hs(Ω)= u L2(Ω): | − | L2(Ω Ω) ,
( ∈ x y N2+s ∈ × )
| − |
endowed with the norm
u(x) u(y)2
u 2 = u2dx+ | − | dxdy. (2.1)
k kHs(Ω) | | x y N+2s
ZΩ ZΩZΩ | − |
Also,wedenote byHs(Ω) the Hilbertspacedefinedasthe closureofC∞(Ω) under
0 0
the norm 2 defined in (2.1). The dual space H−s(Ω) of Hs(Ω) is defined
k·kHs(Ω) 0
in the standard way. For a general reference on analytical properties of fractional
Sobolev spaces, see [27].
The next result is an adaptation of a classical lemma of Lions. For a proof we
refer to [5, Lemma 2.8].
Lemma 2.1. Let N 2 and 2∗ = 2N/(N 2s). If u is bounded in Hs(RN)
≥ s − { n}
and for some R>0 we have
lim sup u (x)2dx=0,
n
n→∞y∈RNZBR(y)| |
then one has u 0 in Lr(RN) for any 2<r <2∗.
n → s
Now we need to introduce some notation. Define
F(z)= z 2Log z 2 for every z C,
| | | | ∈
and as in [11], we define the functions A, B on [0, ) by
∞
s2Log(s2), if 0 s e−3;
A(s)= − ≤ ≤ B(s)=F(s)+A(s). (2.2)
(3s2+4e−3s e−6, if s e−3;
− ≥
FRACTIONAL LOGARITHMIC SCHRO¨DINGER EQUATION 5
Furthermore, let a, b be functions defined by
z z
a(z)= A(z ) and b(z)= B(z ) for z C, z =0. (2.3)
z 2 | | z 2 | | ∈ 6
| | | |
Notice that we have b(z) a(z) = zLog z 2. It follows that A is a nonnegative
− | |
convex and increasing function, and A C1([0,+ )) C2((0,+ )). The Orlicz
space LA(RN) corresponding to A is de∈fined by ∞ ∩ ∞
LA(RN)= u L1 (RN):A(u) L1(RN) ,
∈ loc | | ∈
equipped with the Luxembur(cid:8)g norm (cid:9)
u =inf k >0: A k−1 u(x) dx 1 .
k kLA (cid:26) ZRN | | ≤ (cid:27)
Here as usual L1 (RN) is the space of all loc(cid:0)ally Lebes(cid:1)gue integrable functions. It
loc
is proved in [11, Lemma 2.1] that A is a Young-function which is ∆ -regular and
2
LA(RN), is a separable reflexive Banach space.
k·kLA
We considerthe reflexiveBanachspaceWs(RN)=Hs(RN) LA(RN)equipped
(cid:0) (cid:1) ∩
with the usual norm u = u + u . The following lemma pro-
k kWs(RN) k kHs(RN) k kLA
vides an alternative way of defining the energy space Ws(RN).
Lemma 2.2. Let 0<s<1 and N 2. Then
≥
Ws(RN)= u Hs(RN): u2Log u2 L1(RN)
∈ | | | | ∈
n o
Proof. Oneeasilyverifiesthatforeveryǫ>0,thereexistC >0suchthat B(z)
ǫ
B(z ) C (z 1+ǫ+ z 1+ǫ)z z forall z,z C. Integratingthis inequa|lity o−n
1 ǫ 1 1 1
RN w|it≤h ǫ =|(|2∗ 2|)/2| and| a−pply|ing Ho¨lder in∈equality and Sobolev embeddings
s −
give
γ
B(u) B(v ) dx C u + v u v , (2.4)
ZRN | | | − | | | ≤ (cid:16)k kHs(RN) k kHs(RN)(cid:17) k − kL2
with γ =2∗/2. Thus for u Hs(RN) we get B(u) L1(RN). Lemma 2.2 follows
then from tshe definition of∈the spaces Ws(RN) a|n|d L∈A(RN). (cid:3)
The following lemma is a variant of the Br´ezis-Lieblemma from [9].
Lemma 2.3. Let u be a bounded sequence in Ws(RN) such that u u a.e.
n n
in RN. Then u W{s(R}N) and →
∈
lim u 2Log u 2 u u2Log u u2 dx= u2Log u2dx.
n n n n
n→∞ZRN n| | | | −| − | | − | o ZRN | | | |
Proof. The proof follows along the same lines as [4, Lemma 2.3]. We omit the
details. (cid:3)
It follows from Proposition 1.1.3 in [12] that
W−s(RN)=H−s(RN)+LA′(RN),
wheretheBanachspaceW−s(RN)isequippedwithitsusualnorm. Here,LA′(RN)
is the dual space of LA(RN) (see [11]). It is easy to see that one has the following
chain of continuous embedding: Ws(RN)֒ L2(RN)֒ W−s(RN).
Now we will show that the energyfunct→ional E is o→f class C1 on Ws(RN). First
we need the following lemma.
6 ALEXH.ARDILA
Lemma 2.4. The operator L : u ( ∆)su uLog u2 is continuous from
Ws(RN) to W−s(RN). The image u→nder−L of a−bounde|d|subset of Ws(RN) is
a bounded subset of W−s(RN).
Proof. First, notice that ( ∆)s is continuous from Hs(RN) to H−s(RN). Thus,
usingWs(RN)֒ Hs(RN)−, weobtainthatthe operatoru ( ∆)su iscontinuous
fromWs(RN)t→oW−s(RN). Secondly,foreveryǫ>0,ther→eex−istC >0suchthat
ǫ
b(z) b(z ) C (z ǫ+ z ǫ)z z for all z, z C. Integrating this inequality
1 ǫ 1 1 1
|onRN−withǫ|≤=(2∗| |2)/2|an|d|ap−plyin|gHo¨lderineq∈ualityandSobolevembeddings
s−
we obtain
γ
kb(u)−b(v)kL2(RN) ≤Cku−vkHs(RN) kukHs(RN)+kukHs(RN)
where γ = 2s/(N 2s). Then clearly u b(u(cid:0)) is continuous and bou(cid:1)nded from
Hs(RN) to L2(RN−), then from Ws(RN) t→o W−s(RN). Finally, since u a(u) is
continuousandboundedfromLA(RN)toLA′(RN)(see[11,Lemma2.6]),→itfollows
that the operator u a(u) b(u) = uLog u2 is continuous and bounded from
Ws(RN) to W−s(RN→), and l−emma is p−roved.| | (cid:3)
From Lemma 2.4, we have the following consequence:
Proposition 2.5. The operator E : Ws(RN) R is of class C1 and for u
Ws(RN) the Fr´echet derivative of E in u exists a→nd it is given by ∈
E′(u)=( ∆)su uLog u2 u
− − | | −
Proof. We first show that E is continuous. Notice that
1 1 1
E(u)= ( ∆)s2u2dx+ A(u )dx B(u )dx. (2.5)
2ZRN | − | 2ZRN | | − 2ZRN | |
The first term in the right-hand side of (2.5) is continuous of Hs(RN) R, and
→
it follows from Proposition 6.1(i) in Appendix that the second term is continuous
of LA(RN) R. Moreover, by (2.4) we get that the third term in the right-hand
side of (2.5)→is continuous of Hs(RN) R. Therefore, E C(Ws(RN),R). Now,
direct calculations show that, for u, v →Ws(RN), t ( 1,∈1) (see [11, Proposition
∈ ∈ −
2.7]),
E(u+tv) E(u)
lim − = ( ∆)su uLog u2 u,v .
t→0 t − − | | −
Thus, E is Gˆateaux differentiable. The(cid:10)n, by Lemma 2.4 we see th(cid:11)at E is Fr´echet
differentiable and E′(u)=( ∆)su uLog u2 u. (cid:3)
− − | | −
3. The Cauchy problem
In this section we sketch the proof of the global well-posedness of the Cauchy
Problem for (1.1) in the energy space Ws(RN). The proof of Proposition 1.1
is an adaptation of the proof of [12, Theorem 9.3.4]. So, we will approximate
the logarithmic nonlinearity by a smooth nonlinearity, and as a consequence we
construct a sequence of global solutions of the regularized Cauchy problem in
C(R,Hs(RN)) C1(R,H−s(RN)), then we pass to the limit using standard com-
∩
pactness results, extract a subsequence which converges to the solution of the lim-
iting equation (1.1). Finally, by using special properties of the logarithmic nonlin-
earity we establish uniqueness of the global solution.
FRACTIONAL LOGARITHMIC SCHRO¨DINGER EQUATION 7
First we regularize the logarithmic nonlinearity near the origin. For z C and
m N, we define the functions a and b by ∈
m m
∈
a(z), if z 1; b(z), if z m;
a (z)= | |≥ m and b (z)= | |≤
m (mza(1), if z 1; m (z b(m), if z m,
m | |≤ m m | |≥
where a and b were defined in (2.3). For any fixed m N, we define a family of
regularizednonlinearities in the form g (z)=b (z) a∈ (z), for every z C.
m m m
Inordertoconstructasolutionof (1.1),wesolvefir−st,form N,thereg∈ularized
∈
Cauchy problem
i∂ um ( ∆)sum+g (um)=0. (3.1)
t m
− −
Proposition 3.1. Let 0 < s < 1 and N 2. For any u Hs(RN), there
0
is a unique solution um C(R,Hs(RN)) ≥C1(R,H−s(RN)) o∈f (3.1) such that
∈ ∩
um(0)=u . Furthermore, the conservation of energy and charge holds; that is,
0
(um(t))= (u ) and um(t) 2 = u 2 for all t R,
Em Em 0 k kL2 k 0kL2 ∈
where
1 1 |z|
m(u)= ( ∆)s2u2dx Gm(u)dx, Gm(z)= gm(s)ds.
E 2ZRN | − | − 2ZRN Z0
Proof. Ourproofisinspiredbytheresultsof[16,Section4]. First,sinceg isglob-
m
allyLipschitzcontinuousC→C,oneeasilyverifiesthatkgm(u)−gm(v)kH−s(RN) ≤
C(K)ku−vkHs(RN) provided that kukHs(RN) +kvkHs(RN) ≤ K. Then, from [16,
Proposition 4.1] we see that there exists a weak solution um of (3.1) such that
um L∞(( T ,T ),Hs(RN)) W1,∞(( T ,T ),H−s(RN)),
min max min max
∈ − ∩ −
(um(t)) (u ) and um(t) 2 = u 2 (3.2)
Em ≤Em 0 k kL2 k 0kL2
for all t ( T ,T ), where ( T ,T ) is the maximal existence time
min max min max
∈ − −
interval of um for initial data u .
0
Secondly, we show that there is uniqueness for the problem (3.1). In fact, let I
be an interval containing 0 and let Φ, Ψ L∞(I,Hs(RN)) W1,∞(I,H−s(RN))
∈ ∩
be two solutions of (3.1). It follows that
t
Ψ(t) Φ(t)=i U(t s)(g (Ψ(s)) g (Φ(s)))ds for all t I,
m m
− − − ∈
Z0
whereU(t)=e−it(−∆)s. Sinceg isLipschitzcontinuousL2(RN) L2(RN),there
m
→
exists a constant C >0 such that
t
Ψ(t) Φ(t) 2 C Ψ(s) Φ(s) 2 ds,
k − kL2 ≤ k − kL2
Z0
and therefore the uniqueness follows by Gronwall’s Lemma. Furthermore, since
G (u)dx C u 2 , from (3.2) we get um(t) 2 C u 2 +4 (u )
RN m ≤ k kL2 k kHs(RN) ≤ k 0kL2 Em 0
for all t ( T ,T ). The continuity argument implies that all solutions of
(R3.1) are∈glo−balmainnd umnaifxormly bounded in Hs(RN).
Finally, we prove that the weak solution um of (3.1) satisfies the conservation
of energy. Indeed, fix t R. Let ϕ = um(t ) and let w be the solution of (3.1)
0 0 0
with w(0) = ϕ . By uniq∈ueness, we see that w( t ) = um() on R. From (3.2),
0 0
·− ·
we deduce in particular that
(u ) (ϕ )= (um(t ))
m 0 m 0 m 0
E ≤E E
8 ALEXH.ARDILA
Therefore, we have that both um(t) 2 and (um(t)) are constant on R. The
inclusion um C(R,Hs(RN))k C1(Rk,LH2 −s(RNEm)) follows from conservation laws.
This complete∈s the proof of Pro∩position 3.1. (cid:3)
For the proof of Proposition 1.1, we will use the following lemma.
Lemma3.2. Let0<s<1andN 2. Givenk N,setΩ = x RN : x <k .
k
≥ ∈ ∈ | |
Let {um}m∈N be a bounded sequence in L∞(R,Hs(RN)). I(cid:8)f (um|Ωk)m∈N is(cid:9)a
bounded sequence of W1,∞(R,H−s(Ω )) for k N, then there exists a subsequence,
k
which we still denote by um , and there e∈xist u L∞(R,Hs(RN)), such that
{ }m∈N ∈
the following properties hold:
(i)u W1,∞(R,H−s(Ω )) for every k N.
|Ωk ∈ k ∈
(ii) um(t)⇀u(t) in Hs(RN) as m for every t R.
(iii) For every t R there exists a s→ubs∞equence mj su∈ch that umj(x,t) u(x,t) as
j , for a.e.∈x RN. →
(iv→) u∞m(x,t) u(x∈,t) as m , for a.e. (x,t) RN R.
→ →∞ ∈ ×
Proof. The proof follows a similar argument as in Lemma 9.3.6 of [12] and we do
not repeat here. (cid:3)
Proof of Proposition 1.1. Our proof follows the ideas of Cazenave [12, Theo-
rem 9.3.4]. Applying Proposition 3.1, we see that for every m N there exists
a unique global solution um C(R,Hs(RN)) C1(R,H−s(RN))∈of (3.1), which
∈ ∩
satisfies
(um(t))= (u ) and um(t) 2 = u 2 for all t R, (3.3)
Em Em 0 k kL2 k 0kL2 ∈
where
1 1 1
m(u)= ( ∆)s2u2dx+ Φm(u )dx Ψm(u )dx,
E 2ZRN | − | 2ZRN | | − 2ZRN | |
and the functions Φ and Ψ defined by
m m
1 |z| 1 |z|
Φ (z)= a (s)ds and Ψ (z)= b (s)ds.
m m m m
2 2
Z0 Z0
Itfollowsfrom(3.3)thatumisboundedinL∞(R,L2(RN)). Moreover,wehavethat
thesequenceofapproximatingsolutionsumisboundedinthespaceL∞(R,Hs(RN)).
ItalsofollowsfromtheNLSequation(3.1)thatthesequence ∂ um isboundedin
t |Ωk
the space L∞(R,H−s(Ω )), where Ω = x RN : x <k . Such statements can
k k
∈ | |
be proved along the same lines as the Step 2 of Theorem 9.3.4 in [12]. Therefore,
(cid:8) (cid:9)
we have that um satisfies the assumptions of Lemma 3.2. Let u be the limit
{ }m∈N
of um.
Now we show that the limiting function u L∞(R,Hs(RN)) is a weak solution
∈
ofthelogarithmicNLSequation(1.1). Todoso,wefirstwriteaweakformulationof
theNLSequation(3.1). Indeed,foranytestfunctionψ C∞(RN)andφ C∞(R),
∈ 0 ∈ 0
we have
[ ium,ψ φ′(t) um,( ∆)sψ φ(t)] dt+ g (um)ψφdxdt =0. (3.4)
m
ZR −h i −h − i ZRZRN
Furthermore, since g (z) zLog z 2 pointwise in z C as m + , we apply
m
→ | | ∈ → ∞
the properties (ii)-(iv) of Lemma 3.2 to the integral formulation (3.4) and obtain
FRACTIONAL LOGARITHMIC SCHRO¨DINGER EQUATION 9
the following integral equation (see proof of Step 3 of [12, Theorem 9.3.4])
[ iu,ψ φ′(t) u,( ∆)sψ φ(t)] dt+ uLog u 2ψφdxdt=0. (3.5)
ZR −h i −h − i ZRZRN | |
Inaddition,u(0)=u byproperty(ii)ofLemma3.2. Moreover,itiseasytoseethat
0
u L∞(R,LA(RN)). Therefore,byintegralequation(3.5),u L∞(R,Ws(RN))is
∈ ∈
a weak solution of the logarithmic NLS equation (1.1). In particular,from Lemma
2.4, we deduce that u W1,∞(R,W−s(RN)). Now we show uniqueness of the
solutionintheclassL∞(∈R,Ws(RN)) W1,∞(R,W−s(RN)). Indeed,letuandv be
∩
two solutions of (1.1) in that class. On taking the difference of the two equations
and taking the Ws(RN) W−s(RN) duality product with i(u u), we see that
− −
u v ,u v = uLog u2 vLog v 2 (u v)dx.
t t
h − − i −ℑZRN (cid:16) | | − | | (cid:17) −
Thus, from [12, Lemma 9.3.5] we obtain
t
u(t) v(t) 2 8 u(s) v(s) 2 ds.
k − kL2 ≤ k − kL2
Z0
Therefore, the uniqueness of a solution follows by Gronwall’s Lemma. Finally,
the conservation of charge and energy, and the continuity of the solution u
C(R,Ws(RN)) C1(R,W−s(RN)) in time t follow from the arguments identica∈l
∩
to the case of the classical logarithmic NLS equation (see proof of Step 4 of [12,
Theorem 9.3.4]). This finishes the proposition. (cid:3)
4. Variational Analysis
The aim of this section is to prove Theorem 1.2. First we recall the fractional
logarithmic Sobolev inequality. For a proof we refer to [17].
Lemma 4.1. Let f be any function in Hs(RN) and α>0. Then
f(x)2 N sΓ(N) α2
ZRN |f(x)|2Log |kfk2L|2 !dx+ N + s Logα+Log Γ(2Ns2) !kfk2L2 ≤ πsk(−∆)s2fk2L2.
(4.1)
Lemma 4.2. Let ω R. Then, the quantity d(ω) is positive and satisfies
∈
1 sΓ(N)
d(ω) 2 πN2eω+N. (4.2)
≥ 2 Γ(N) !
2s
Proof. Let u Ws(RN) 0 be such that I (u) = 0. Using the fractional loga-
ω
rithmic Sobol∈ev inequalit\y{wi}th α=πs2, we see that
sΓ(N)
ω+N(1+Log(√π))+Log 2 u 2 Log u 2 u 2 .
Γ(N) !k kL2 ≤ k kL2 k kL2
2s (cid:16) (cid:17)
Thus, by the definition of d(ω) given in (1.7), we get (4.2). (cid:3)
Lemma 4.3. Let ω R and 0 < s < 1. If u is a minimizing sequence of
n
problem (1.7), then th∈ere is a subsequence u {an}d a sequence y RN such
{ jn} { n} ⊂
that
v (x):=u (x+y )
n jn n
converges weakly in Ws(RN) to a function ϕ = 0. Moreover, v converges to ϕ
n
a.e and in Lq (RN) for every q [2,2∗). 6 { }
loc ∈ s
10 ALEXH.ARDILA
Proof. Let u Ws(RN) be a minimizing sequence for d(ω), then the sequence
n
u is bou{nd}ed⊆in Ws(RN). Indeed, it is clear that the sequence u 2 is
{ n} k nkL2
bounded. Moreover, using the fractional logarithmic Sobolev inequality and re-
calling that I (u )=0, we obtain
ω n
1− απs2 (−∆)2sun 2L2 ≤Log e−s(αω+NN/s)ΓΓ((NN//22)s) kunk2L2 kunk2L2.
(cid:18) (cid:19) (cid:20)(cid:18) (cid:19) (cid:21)
Taking α > 0 s(cid:13)(cid:13)ufficiently(cid:13)(cid:13)small, we see that k(−∆)2sunk2L2 is bounded, so the
sequence u is bounded in Hs(RN). Then, using I (u )=0 again, and (2.4) we
n ω n
{ }
obtain
A(u )dx B(u )dx+ ω u 2 C,
ZRN | n| ≤ZRN | n| | |k nkL2 ≤
which implies, by (6.1) in the Appendix, that the sequence u is bounded in
n
Ws(RN). Furthermore, since Ws(RN) is a reflexive Banach{spa}ce, there is v
Ws(RN) such that, up to a subsequence, u ⇀v weakly in Ws(RN). ∈
n
Ontheotherhand,let2<q <2∗and0<δ <1. Noticethat u 2Log u 2dx=
s RN | n| | n|
k(−∆)2sunk2L2 +ωkunk2L2 ≥−M(ω) for sufficiently large n, whRere M(ω) is a posi-
tive constant depending only on ω. Arguing as in the proof of Lemma 3.3 in [11],
it is easy to see that
2d(ω) (δ−(q−2) (q 2)Logδ2 −1) u qdx Logδ2 −1M(ω).
n
≤ − − ZRN | | −
(cid:2) (cid:3) (cid:2) (cid:3)
Wenowchooseδ suchthat Logδ2 −1M(ω)=d(ω)((q 2)/(q+2)). Easycom-
− −
putations permit us to obtain
(cid:2) (cid:3)
d(ω) q+6
u qdx eM(ω)(q+2)/2d(ω)+ 2d(ω) d(ω). (4.3)
n
ZRN | | (cid:18) M(ω)(q+2)(cid:19)≥ (cid:18)q+2(cid:19)≥
Combining (4.3) and Lemma 2.1 implies
sup u 2dx ǫ>0,
n
y∈RNZB1(y)| | ≥
In this case we can choose y RN such that
n
{ }⊂
u ( +y )2dx ǫ′,
n n
| · | ≥
ZB1(0)
where0<ǫ′ <ǫ,andhence,duetothecompactnessoftheembeddingHs (RN)֒
L2 (RN),wededucethatthetranslatedsequencev :=u ( +y )hasalwoceaklim→it
ϕloinc Hs(RN) that is not identically zero. Also, it fonllowsnth·at nv convergesto ϕ
n
stronglyinLq (RN)foranyq [2,2∗). Therefore,weinferthat{,af}teratranslation
ifnecessary, louc convergeswe∈aklyins Ws(RN)anda.e. toafunctionϕ=0. Hence
n
the result is{esta}blished. 6 (cid:3)
Proof of Theorem 1.2. The proof follows basically the same idea as the proof
of [4, Proposition 1.3 and Lemma 3.1] (see also [1]). Let u Ws(RN) be a
n
minimizing sequence for d(ω). From Lemma 4.3, there exi{st ϕ} ⊆Ws(RN) 0
such that, v := u ( +y ) ⇀ ϕ weakly in Ws(RN) and v co∈nverges to \ϕ{a.}e
and in Lq (RnN) fojrne·verynq [2,2∗). { n}
loc ∈ s
Now we prove that ϕ , that is, I (ϕ)=0 and S (ϕ)=d(ω). First, assume
ω ω ω
∈G
by contradiction that I (ϕ) < 0. By elementary computations, we can see that
ω