A VARIATIONAL ANALYSIS OF A GAUGED NONLINEARSCHR ¨ ODINGER EQUATION
ALESSIO POMPONIO
1
AND DAVID RUIZ
2
A
BSTRACT
. This paper is motivated by a gauged Schr¨odinger equation in dimension 2 including the socalled ChernSimons term. The study of radial stationarystates leads to the nonlocal problem:
−
∆
u
(
x
) +
ω
+
h
2
(

x

)

x

2
+
+
∞
x

h
(
s
)
su
2
(
s
)
ds
u
(
x
) =

u
(
x
)

p
−
1
u
(
x
)
,
where
h
(
r
) = 12
r
0
su
2
(
s
)
ds.
This problem is the EulerLagrange equation of a certain energy functional.In this paper the study of the global behavior of such functional is completed.We show that for
p
∈
(1
,
3)
, the functional may be bounded from below or not,depending on
ω
. Quite surprisingly, the threshold value for
ω
is explicit. Fromthis study we prove existence and nonexistence of positive solutions.
1. I
NTRODUCTION
In this paper we are concerned with a planar gauged Nonlinear Schr¨odingerEquation:(1)
iD
0
φ
+ (
D
1
D
1
+
D
2
D
2
)
φ
+

φ

p
−
1
φ
= 0
.
Here
t
∈
R
,
x
= (
x
1
,x
2
)
∈
R
2
,
φ
:
R
×
R
2
→
C
is the scalarﬁeld,
A
µ
:
R
×
R
2
→
R
are the components of the gauge potential and
D
µ
=
∂
µ
+
iA
µ
is the covariantderivative (
µ
= 0
,
1
,
2
).Theclassicalequationforthegaugepotential
A
µ
istheMaxwellequation. However, the modiﬁed gauge ﬁeld equation proposes to include the socalled ChernSimons term into that equation (see for instance [20, Chapter 1]):(2)
∂
µ
F
µν
+ 12
κǫ
ναβ
F
αβ
=
j
ν
,
with
F
µν
=
∂
µ
A
ν
−
∂
ν
A
µ
.
In the above equation,
κ
is a parameter that measures the strength of the ChernSimons term. As usual,
ǫ
ναβ
is the LeviCivita tensor, and superindices are relatedto the Minkowski metric with signature
(1
,
−
1
,
−
1)
. Finally,
j
µ
is the conservedmatter current,
j
0
=

φ

2
, j
i
= 2Im
¯
φD
i
φ
.
2010
Mathematics Subject Classiﬁcation.
35J20, 35Q55.
Key words and phrases.
Gauged Schr¨odinger Equations, ChernSimons theory, Variational methods,concentration compactness.A.P. is supported by M.I.U.R.  P.R.I.N. “Metodi variazionali e topologici nello studio di fenomeninon lineari”, by GNAMPA Project “Metodi Variazionali e Problemi Ellittici Non Lineari” and byFRA2011 “Equazioni ellittiche di tipo BornInfeld”. D.R. is supported by the Spanish Ministry of Science and Innovation under Grant MTM201126717 and by J. Andalucia (FQM 116).
1
a r X i v : 1 3 0 6 . 2 0 5 1 v 1 [ m a t h . A P ] 9 J u n 2 0 1 3
2 POMPONIO AND RUIZ
At low energies, the Maxwell term becomes negligible and can be dropped,giving rise to:(3)
12
κǫ
ναβ
F
αβ
=
j
ν
.
See [7,8,12–14] for the discussion above.
For the sake of simplicity, let us ﬁx
κ
= 2
. Equations (1) and (3) lead us to the
problem:(4)
iD
0
φ
+ (
D
1
D
1
+
D
2
D
2
)
φ
+

φ

p
−
1
φ
= 0
,∂
0
A
1
−
∂
1
A
0
= Im(¯
φD
2
φ
)
,∂
0
A
2
−
∂
2
A
0
=
−
Im(¯
φD
1
φ
)
,∂
1
A
2
−
∂
2
A
1
=
12

φ

2
.
As usual in ChernSimons theory, problem (4) is invariant under gauge trans
formation,(5)
φ
→
φe
iχ
, A
µ
→
A
µ
−
∂
µ
χ,
for any arbitrary
C
∞
function
χ
.This model was ﬁrst proposed and studied in [12–14], and sometimes has re
ceived the name of ChernSimonsSchr¨odinger equation. The initial value problem, as well as global existence and blowup, has been addressed in [2,9,11] for
the case
p
= 3
.The existence of stationary states for (4) and general
p >
1
has been studiedrecently in [4] (with respect to that paper, our notation interchanges the indices
1
and
2
). By using the ansatz:
φ
(
t,x
) =
u
(

x

)
e
iωt
, A
0
(
x
) =
A
0
(

x

)
,A
1
(
t,x
) =
−
x
2

x

2
h
(

x

)
, A
2
(
t,x
) =
x
1

x

2
h
(

x

)
,
in [4] it is found that
u
solves the equation::(6)
−
∆
u
(
x
)+
ω
+
ξ
+
h
2
(

x

)

x

2
+
+
∞
x

h
(
s
)
s u
2
(
s
)
ds
u
(
x
) =

u
(
x
)

p
−
1
u
(
x
)
, x
∈
R
2
,
where
h
(
r
) = 12
r
0
su
2
(
s
)
ds.
Here
ξ
in
R
is an integration constant of
A
0
, which takes the form:
A
0
(
r
) =
ξ
+
+
∞
r
h
(
s
)
s u
2
(
s
)
ds.
Observe that (6) is a nonlocal equation. Moreover, in [4] it is shown that (6) is
indeed the EulerLagrange equation of the energy functional:
I
ω
+
ξ
:
H
1
r
(
R
2
)
→
R
,
deﬁned as
I
ω
+
ξ
(
u
) = 12
R
2
∇
u
(
x
)

2
+ (
ω
+
ξ
)
u
2
(
x
)
dx
+ 18
R
2
u
2
(
x
)

x

2

x

0
su
2
(
s
)
ds
2
dx
−
1
p
+ 1
R
2

u
(
x
)

p
+1
dx.
Here
H
1
r
(
R
2
)
denotes the Sobolev space of radially symmetric functions. It is important to observe that the energy functional
I
ω
+
ξ
presents a competition between
A VARIATIONAL ANALYSIS OF A GAUGED NONLINEAR SCHR ¨ODINGER EQUATION 3
the nonlocal term and the local nonlinearity. The study of the behavior of thefunctional under this competition is one of the main motivations of this paper.Given a stationary solution, and taking
χ
=
ct
in the gauge invariance (5), we
obtain another stationary solution; the functions
u
(
x
)
,
A
1
(
x
)
,
A
2
(
x
)
are preserved,and
ω
→
ω
+
c, A
0
(
x
)
→
A
0
(
x
)
−
c
Therefore, the constant
ω
+
ξ
is a gauge invariant of the stationary solutions of the problem. By the above discussion we can take
ξ
= 0
in what follows, that is,
lim

x
→
+
∞
A
0
(
x
) = 0
,
which was indeed assumed in [2,14].
For
p >
3
, it is shown in [4] that
I
ω
is unbounded from below, so it exhibitsa mountainpass geometry. In a certain sense, in this case the local nonlinearitydominates the nonlocal term. However the existence of a solution is not so direct,since for
p
∈
(3
,
5)
the (PS) property is not known to hold. This problem is bypassed in [4] by using a constrained minimization taking into account the Nehari
and Pohozaev identities, in the spirit of [17]. Moreover, inﬁnitely many solutions
have been found in [10] for
p >
5
(possibly signchanging).A special case in the above equation is
p
= 3
: in this case, static solutions can befound by passing to a selfdual equation, which leads to a Liouville equation thatcan be solved explicitly. Those are the unique positive solutions, as proved in [4].
For more information on the selfdual equations, see [5,14,20].
Incase
p
∈
(1
,
3)
, solutionsarefoundin[4]asminimizersona
L
2
sphere. Therefore, thevalue
ω
comesoutasaLagrangemultiplier, anditisnotcontrolled. Moreover, the global behavior of the energy functional
I
ω
is not studied.The main purpose of this paper is to study whether
I
ω
is bounded from belowor not for
p
∈
(1
,
3)
. In this case, the nonlocal term prevails over the local nonlinearity, in a certain sense. As we shall see, the situation is quite rich and unexpecteda priori, and very different from the usual Nonlinear Schr¨odinger Equation. Thissituation differs also from the Schr¨odingerPoisson problem (see [17]), which is
another problem presenting a competition between local and nonlocal nonlinearities.We shall prove the existence of a threshold value
ω
0
such that
I
ω
is boundedfrom below if
ω
ω
0
, and it is not for
ω
∈
(0
,ω
0
)
. But, in our opinion, what ismost surprising is that
ω
0
has an explicit expression, namely:(7)
ω
0
= 3
−
p
3 +
p
3
p
−
12(3
−
p
)
2
23
−
p
m
2
(3 +
p
)
p
−
1
−
p
−
12(3
−
p
)
,
with
m
=
+
∞−∞
2
p
+ 1 cosh
2
p
−
12
r
21
−
p
dr.
Let us give an idea of the proofs. It is not difﬁcult to show that
I
ω
is coercivewhen the problem is posed on a bounded domain. So, there exists a minimizer
u
n
on the ball
B
(0
,n
)
with Dirichlet boundary conditions. To prove boundedness of
u
n
, the problem is the possible loss of mass at inﬁnity as
n
→
+
∞
. The core of ourproofs is a detailed study of the behavior of those masses. We are able to showthat, if unbounded, the sequence
u
n
behaves as a soliton, if
u
n
is interpreted as afunction of a single real variable. The proof uses a careful study of the level sets of
u
n
, which take into account the effect of the nonlocal term. Then, the energy functional
I
ω
admits a natural approximation through a convenient limit functional.
4 POMPONIO AND RUIZ
Finally, the solutions of that limit functional, and their energy, can be found explicitly, so we can ﬁnd
ω
0
. See Section 2 for an heuristic explanation of the proof and a derivation of the limit functional.Regarding the existence of solutions, a priori, the global minimizer could correspond to the zero solution. And indeed this is the case for large
ω
. Instead, weshow that
inf
I
ω
<
0
if
ω > ω
0
is close to the threshold value. Therefore, the globalminimizerisnottrivial, andcorrespondstoapositivesolution. Themountainpasstheorem will provide the existence of a second positive solution.If
ω < ω
0
,
I
ω
is unbounded from below, and hence the geometric assumptionsof the mountainpass theorem are satisﬁed. However, the boundedness of (PS)sequences seems to be a hard question in this case. Solutions are found for almostallvaluesof
ω
∈
(0
,ω
0
)
,byusingthewellknownmonotonicitytrickofStruwe[19]
(see also [15]).
Our main results are the following:
Theorem 1.1.
For
ω
0
as given in
(7)
, there holds:(i) if
ω
∈
(0
,ω
0
)
, then
I
ω
is unbounded from below;(ii) if
ω
=
ω
0
, then
I
ω
0
is bounded from below, not coercive and
inf
I
ω
0
<
0
;(iii) if
ω > ω
0
, then
I
ω
is bounded from below and coercive.
Regarding the existence of solutions, we obtain the following result:
Theorem 1.2.
Consider
(6)
with
ξ
= 0
. There exist
¯
ω >
˜
ω > ω
0
such that:(i) if
ω >
¯
ω
, then
(6)
has no solutions different from zero;(ii) if
ω
∈
(
ω
0
,
˜
ω
)
, then
(6)
admits at least two positive solutions: one of them is a global minimizer for
I
ω
and the other is a mountainpass solution;(iii) for almost every
ω
∈
(0
,ω
0
)
(6)
admits a positive solution.
The rest of the paper is organized as follows. Section 2 is devoted to some preliminary results. Moreover, we give a heuristic presentation of our proofs, whichmotivates the deﬁnition of the limit functional. This limit functional is studied indetail in Section 3. Finally, in Section 4 we prove Theorems 1.1 and 1.2.
Acknowledgement.
This work has been partially carried out during a stay of A.P.in Granada. He would like to express his deep gratitude to the Departamento deAn´alisis Matem´atico for the support and warm hospitality.2. P
RELIMINARIES
Let us ﬁrst ﬁx some notations. We denote by
H
1
r
(
R
2
)
the Sobolev space of radially symmetric functions, and
·
its usual norm. Other norms, like Lebesguenorms, will be indicated with a subscript. In particular,
·
H
1
(
R
)
,
·
H
1
(
a,b
)
areused to indicate the norms of the Sobolev spaces of dimension
1
. If nothing isspeciﬁed, strong and weak convergence of sequences of functions are assumed inthe space
H
1
(
R
2
)
.In our estimates, we will frequently denote by
C >
0
,
c >
0
ﬁxed constants,that may change from line to line, but are always independent of the variableunder consideration. We also use the notations
O
(1)
,o
(1)
,O
(
ε
)
,o
(
ε
)
to describethe asymptotic behaviors of quantities in a standard way. Finally the letters
x
,
y
indicate twodimensional variables and
r
,
s
denote onedimensional variables.Let us start with the following proposition, proved in [4]:
Proposition 2.1.
I
ω
is a
C
1
functional, and its critical points correspond to classicalsolutions of
(6)
.
A VARIATIONAL ANALYSIS OF A GAUGED NONLINEAR SCHR ¨ODINGER EQUATION 5
Next result deals with the behavior of
I
ω
under weak limits in
H
1
r
(
R
2
)
. Evenif it is not explicitly stated in this form, Proposition 2.2 follows easily from [4,
Lemma3.2]andthecompactnessoftheembedding
H
1
r
(
R
2
)
→
L
q
(
R
2
)
,
q
∈
(2
,
+
∞
)
(see [18]).
Proposition 2.2.
If
u
n
⇀ u
, then
R
2
u
2
n
(
x
)

x

2

x

0
su
2
n
(
s
)
ds
2
dx
→
R
2
u
2
(
x
)

x

2

x

0
su
2
(
s
)
ds
2
dx.
Inparticular,
I
ω
isweaklowersemicontinuous. Moreover, if
u
n
⇀ u
then
I
′
ω
(
u
n
)(
ϕ
)
→
I
′
ω
(
u
)(
ϕ
)
for all
ϕ
∈
H
1
r
(
R
2
)
.
To ﬁnish the account of preliminaries, we now state an inequality which willprove to be fundamental in our analysis. This inequality is proved in [4], wherealso the maximizers are found.
Proposition 2.3.
For any
u
∈
H
1
r
(
R
2
)
,
(8)
R
2

u
(
x
)

4
dx
2
R
2
∇
u
(
x
)

2
dx
1
/
2
R
2
u
2

x

2

x

0
su
2
(
s
)
ds
2
dx
1
/
2
.
As commented in the introduction, this paper is concerned with boundednessfrom below of
I
ω
. Let us give a rough idea of the arguments of our proof. First of all, consider
u
(
r
)
a ﬁxed function, and deﬁne
u
ρ
(
r
) =
u
(
r
−
ρ
)
. Let us now estimate
I
ω
(
u
ρ
)
as
ρ
→
+
∞
.
(2
π
)
−
1
I
ω
(
u
ρ
) = 12
+
∞−
ρ
(

u
′

2
+
ωu
2
)(
r
+
ρ
)
dr
+ 18
∞−
ρ
u
2
(
r
)
r
+
ρ
r
−
ρ
(
s
+
ρ
)
u
2
(
s
)
ds
2
dr
−
1
p
+ 1
∞−
ρ

u

p
+1
(
r
+
ρ
)
dr.
We estimate the above expression by simply replacing the expressions
(
r
+
ρ
)
,
(
s
+
ρ
)
with the constant
ρ
:
(2
π
)
−
1
I
ω
(
u
)
∼
ρ
12
+
∞−∞
(

u

′
2
+
ωu
2
)
dr
+ 18
+
∞−∞
u
2
(
r
)
r
−∞
u
2
(
s
)
ds
2
dr
−
1
p
+ 1
+
∞−∞

u

p
+1
dr
=
ρ
12
+
∞−∞
(

u

′
2
+
ωu
2
)
dr
+ 124
+
∞−∞
u
2
dr
3
−
1
p
+ 1
+
∞−∞

u

p
+1
dr
.
This estimate will be made rigorous in Lemma 4.1. Therefore, it is natural to
deﬁne the limit functional
J
ω
:
H
1
(
R
)
→
R
,
J
ω
(
u
) = 12
+
∞−∞

u
′

2
+
ωu
2
dr
+ 124
+
∞−∞
u
2
dr
3
−
1
p
+ 1
+
∞−∞

u

p
+1
dr.
As a consequence of the above argument, if
J
ω
attains negative values, then
I
ω
will be unbounded from below.The reverse is also true, but the proof is more delicate. We will show that if
u
n
is unbounded in
H
1
r
(
R
2
)
and
I
ω
(
u
n
)
is bounded from above, then somehow
u
n
contains a certain mass spreading to inﬁnity, as
u
ρ
does. This will be made explicitin Proposition 4.2. But this will lead us to a contradiction if
J
ω
is positive on thatmass. The proof of this argument is however far from trivial, and is the core of thispaper.