Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 82, pp. 1–12.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
SOLITARY WAVES FOR THE COUPLED NONLINEAR
KLEIN-GORDON AND BORN-INFELD TYPE EQUATIONS
FEIZHI WANG
Abstract. In this article we study the existence of solutions for a nonlinear
Klein-Gordon-Maxwell equation coupled with a Born-Infeld equation.
1. Introduction
It is well known that the gauge potential (φ, A) can be coupled to a complex order
parameter ψ through the minimal coupling rule; that is the formal substitution
∂
∂
7→
+ ieφ,
∂t
∂t
∇ 7→ ∇ − ieA,
where e is the electric charge, A : R3 × R → R3 is a magnetic vector potential and
φ : R3 ×R → R is an electric potential. Therefore, in a flat Minkowskian space-time
with metric (gµν ) = diag[1, −1, −1, −1], we can define the Klein-Gordon-Maxwell
Lagrangian density
1
1 ∂ψ
LKGM = |
+ ieφψ|2 − |∇ψ − ieA|2 − m2 |ψ|2 + |ψ|q ,
2 ∂t
q
where m ≥ 0 represents the mass of the charged field. The total action of the
system is thus given by
ZZ
S=
(LKGM + Lemf ) dx dt,
(1.1)
where Lemf is the Lagrangian density of the electro-magnetic field. In the BornInfeld theory (see [8]), with a suitable choice of constants, Lemf can be written
as
r
b2 1
1 − 1 − 2 (|E|2 − |B|2 ) ,
Lemf = LBI :=
4π
b
where b is the so-called Born-Infeld parameter, b 1. By the Maxwell equations,
∂A
E = −∇φ −
∂t
2000 Mathematics Subject Classification. 35J15.
Key words and phrases. Nonlinear Klein-Gordon; Born-Infeld type equation;
electrostatic solitary wave; critical points.
c
2012
Texas State University - San Marcos.
Submitted December 22, 2011. Published May 23, 2012.
Supported by grant ZR2011AL009 from the Shandong Province Natural Science Foundation.
1
2
F. WANG
EJDE-2012/82
is the electric field, and
B=∇×A
is the magnetic induction field. If, as in [4], we consider the electrostatic solitary
wave:
ψ(x, t) = u(x)e−iωt , A = 0, φ = φ(x),
where u : R3 → R and ω ∈ R, then the total action in (1.1) takes the form
Z
Z
1
1
2
|Du| dx +
m2 − (eφ − ω)2 u2 dx
FBI (u, φ) =
2 R3
2 R3
r
Z
Z 1
1
b2
q
−
|u| dx −
1 − 1 − 2 |∇φ|2 dx.
q R3
4π R3
b
(1.2)
The critical point (u, φ) of FBI satisfies the Euler-Lagrange equations associted to
(1.2). By standard calculations, we obtain:
−∆u + [m2 − (φ − ω)2 ]u = |u|q−2 u,
∇φ
∇· q
1−
= 4π(φ − ω)u2 ,
1
2
b2 |∇φ|
in R3 ,
in R3 ,
(1.3)
where we have taken e = 1. We can see that the sign ω is not relevant for the
existence of solutions for problem (1.3). In fact, if (u, φ) is a solution of (1.3)
with ω, then (u, −φ) is also a solution corresponding to −ω. So, without loss of
generality, we can assume ω > 0.
As we know, a large number of works have been devoted to the problem like
(1.3). In the following we review some assumptions and the corresponding results.
In [2, 3, 4, 5, 6, 7, 9, 10, 15], the authors consider the first-order expansion of
the second formula of (1.3) for b → +∞. Therefore (1.3) becomes
−∆u + [m2 − (φ − ω)2 ]u = |u|q−2 u,
∆φ = 4π(φ − ω)u2 ,
in R3 ,
in R3 .
(1.4)
About the problem (1.4), the pioneering work is given by Benci and Fortunato
[4]. They showed that (1.4) has infinitely many solutions when q ∈ (4, 6) and
0 < ω < m. In [10] d’Aprile and Mugnai proved the existence of nontrivial solutions
of (1.4) whenever q ∈ (2, 4] and
q−2 2
m > ω2 .
2
d’Aprile and Mugnai [9] also showed that (1.4) has no nontrivial solutions when
q ≥ 6 and 0 < ω ≤ m or q ≤ 2. Recently, in [2], under the following conditions:
(q − 2)(4 − q)m2 > ω 2 ,
m > ω > 0,
p ∈ (2, 3),
p ∈ [3, 6),
Azzollini, Pisani and Pomponio showed that (1.4) admits a nontrivial solution. It
is easy to see that (p − 2)(4 − p) > (p − 2)/2 for p ∈ (2, 3].
In [11, 12, 14], the authors consider the second-order expansion of the second
formula of (1.3) for b → +∞. Therefore (1.3) becomes
−∆u + [m2 − (φ − ω)2 ]u = |u|q−2 u,
∆φ + β2 ∆4 φ = 4π(φ − ω)u2 ,
in R3 ,
in R3 ,
(1.5)
EJDE-2012/82
SOLITARY WAVES
3
where β2 = 1/(2b2 ) → 0 and ∆4 φ = D(|Dφ|2 Dφ). In [12], Fortunato, Orsina and
Pisani showed the existence of electrostatic solutions with finite energy, while in
[11] d’Avenia and Pisani proved that (1.5) has infinitely many solutions, provided
that 4 < q < 6 and 0 < ω < m. In [14] Mugnai established the same results under
the following assumptions: 4 ≤ q < 6 and 0 < ω < m or 2 < q < 4 and
q−2 2
m > ω2 .
2
Recently, Yu [18] studied the original Born-Infeld equations, i.e. (1.3). He proved
the existence of the least-action solitary waves in both bounded smooth domain case
and R3 case whenever q ∈ (2, 6) and
q−2 2
m > ω2 .
q
In the present paper we consider the nonlinear Klein-Gordon equations coupled
with the N -th order expansion of the second formula of (1.3) for b → +∞:
−∆u + [m2 − (φ − ω)2 ]u = |u|q−2 u,
N
X
(βk ∆2k φ) = 4π(φ − ω)u2 ,
in R3 ,
in R3 ,
(1.6)
k=1
1
and ∆2k φ = D(|Dφ|2k−2 Dφ), for k =
where β1 = 1, βk = 1·3·5...(2k−3)
2k−1 (k−1)! b2(k−1)
2, 3, . . . , N .
It is well-known that H 1 (R3 ) is the usual Sobolev space endowed with the norm
Z
1/2
kukH 1 (R3 ) =
[|Du|2 + u2 ] dx
R3
(see [1], [17, Theorem 1.8]). D (R ) denotes the completion of C0∞ (R3 , R) with
respect to the norm
1/(2N )
1/2 Z
Z
2
|Dφ|2N dx
.
|Dφ| dx
+
kφkDN (R3 ) =
N
R3
3
R3
By a solution (u, φ) of (1.6), we understand (u, φ) ∈ H 1 (R3 ) × DN (R3 ) satisfying
(1.6) in the weak sense. Obviously, (u, φ) = (0, 0) is a trivial solution of (1.6). We
define a functional FN : H 1 (R3 ) × DN (R3 ) → R by
Z
N
1
1
1 X 1
1
FN (u, φ) =
|Du|2 −
βk |Dφ|2k + (m2 − (φ − ω)2 )u2 − |u|q dx .
4π
2k
2
q
R3 2
k=1
It is easy to see that FN ∈ C 1 (H 1 (R3 ) × DN (R3 ), R). Therefore solutions of (1.6)
correspond to critical points of the functional FN . Next we give our main result.
Theorem 1.1. Problem (1.6) has at least a nontrivial solution (u, φ) ∈ H 1 (R3 ) ×
DN (R3 ), provided one of the following conditions is satisfied
(i) q ∈ (3, 6) and m > ω > 0.
(ii) q ∈ (2, 3] and (q − 2)(4 − q)m2 > ω 2 > 0.
R
1
3
Set |u|q := { R3 |u|q dx}1/q for 1 < q < ∞.
We say that {un } ⊂ H (R ) is a
1
1
3
Palais-Smale sequence for Φ ∈ C H (R ), R at level c ∈ R (the (P S)c -sequence
for short), if and only if {un } satisfies Φ(un ) → c and Φ0 (un ) → 0 as n → ∞.
To find the critical points of the functional FN (u, φ) we will overcome two difficulties. The first difficulty is that FN (u, φ) is strongly indefinite (unbounded both
4
F. WANG
EJDE-2012/82
from below and from above on infinite dimensional subspaces). To avoid this difficulty, we use the reduction method just like in [12, 11, 14]. The reduction method
consists in reducing the study of FN (u, φ) to the study of a functional J(u) in the
only variable u. The second difficulty is that the embedding of H 1 (R3 ) into Lq (R3 )
is not compact, where 2 < q < 2∗ (= 6). So J(u) does not in general satisfy the
Palais-Smale condition. We will study J(u) in Hr1 (R3 ), where
Hr1 (R3 ) = {u ∈ H 1 (R3 ) : u(x) = u(|x|)}.
By the Principle of symmetric criticality (see [16] or [17, Theorem 1.28]), a critical
point u ∈ Hr1 (R3 ) for J(u) is also a critical point in H 1 (R3 ). We construct a
bounded (P S)c -sequence following the methods of Jeanjean [13]. Then there exists
a subsequence of {un } which converges strongly in Hr1 (R3 ).
This paper is organized as follows: in Section 2, we make some preliminaries; in
Section 3, we obtain that the solutions of (1.6) must verify some suitable Pohožaev
identity; in Section 4, we give the proof of Theorem 1.1.
2. Preliminaries
In the following we give some lemmas, whose similar proofs can be founded in
[9, 11, 14].
Lemma 2.1. For every u ∈ H 1 (R3 ) there is a unique φ = Φ(u) ∈ DN (R3 ) which
solves
N
X
(βk ∆2k φ) = 4π(φ − ω)u2 .
(2.1)
k=1
Lemma 2.2. For any u ∈ H 1 (R3 ), on the set {x ∈ R3 : u(x) 6= 0},
0 ≤ Φ(u) ≤ ω.
Proof. Set Φ− = min{Φ, 0}. Multiplying (2.1) by Φ− , we have
Z
Z
N
Z
1 X
−
βk
|DΦ− |2k dx =
(Φ− )2 u2 dx − ω
Φ− u2 dx ≥ 0.
4π
3
3
3
R
R
R
k=1
So we obtain DΦ− ≡ 0. Hence, Φ ≥ 0.
When we multiply (2.1) by (Φ(u) − ω)+ = max{Φ(u) − ω, 0}, we obtain
Z
Z
N
1 X
(Φ(u) − ω)2 u2 dx = −
βk
|DΦ(u)|2k dx ≥ 0,
4π
Φ(u)≥ω
Φ(u)≥ω
k=1
+
so that (Φ(u) − ω) = 0 for u 6= 0. Hence Φ(u) ≤ ω.
Lemma 2.3. The pair (u, φ) ∈ H 1 (R3 ) × DN (R3 ) is a solution of (1.6) if and only
if u is a critical point of
Z h
N
1
1 X 1
2
JN (u) := FN (u, Φ(u)) =
|Du| −
βk |DΦ(u)|2k
2
4π
2k
3
R
k=1
Z
i
1 2
1
+ (m − (Φ(u) − ω)2 )u2 −
|u|q dx
2
q R3
and φ = Φ(u).
EJDE-2012/82
SOLITARY WAVES
5
The functional of (1.6) is
N
1 X 1
βk |Dφ|2k
4π
2k
R3 2
k=1
Z
i
1
1
+ (m2 − (φ − ω)2 )u2 −
|u|q dx.
2
q R3
Z
FN (u, φ) =
h1
|Du|2 −
From Lemma 2.1, for fixed u ∈ H 1 (R3 ), we have
−
Z
Z
N
Z
1 X
βk
|DΦ(u)|2k dx =
Φ2 (u)u2 dx − ω
Φ(u)u2 dx,
4π
R3
R3
R3
k=1
where Φ(u) appears in Lemma 2.1. Then
JN (u) = FN (u, Φ(u))
Z
Z
Z
1
ω
1
|Du|2 dx + (m2 − ω 2 )
Φ(u)u2 dx
u2 dx +
=
2 R3
2
2 R3
R3
Z
N
1Z
1 X k − 1
2k
+
βk
|DΦ(u)| dx −
|u|q dx.
4π
2k
q R3
R3
k=2
By the definition of JN (u), we have
Z
Z
Z
0
hJN
(u), ui =
|Du|2 dx + (m2 − ω 2 )
u2 dx −
Φ2 (u)u2 dx
3
3
3
R
R
R
Z
Z
2
q
+ 2ω
Φ(u)u dx −
|u| dx.
R3
R3
From Lemmas 2.1 and 2.3, to obtain a solution of (1.6), we need only to find
a critical point of JN in H 1 (R3 ). Note that the functional JN depends only on u.
Set
Hr1 (R3 ) = {u ∈ H 1 (R3 ) : u(x) = u(|x|)}.
By standard arguments (Principle of symmetric criticality) one sees that a critical
point u ∈ Hr1 (R3 ) for the functional JN in Hr1 (R3 ) is also a critical point for JN in
H 1 (R3 ).
3. The Pohožaev identity
In this section we obtain that the solutions of (1.6) must verify some suitable
Pohožaev identity, as was proved in [9], which provides necessary conditions to
prove the existence of nontrivial solutions.
2
2k
Lemma 3.1. Let u ∈ Hloc
(Rn ), φ ∈ Hloc
(Rn ) and a, b ≥ 0. Then, for any ball
n
BR = {x ∈ R : |x| ≤ R > 0}, the following equalities hold:
Z
−∆uhx, Dui dx
BR
Z
Z
Z
(3.1)
2−n
1
R
=
|Du|2 dx −
hx, Dui2 dσ +
|Du|2 dσ;
2
R ∂BR
2 ∂BR
BR
6
F. WANG
EJDE-2012/82
Z
(a + bφ)φuhx, Dui dx
Z
a
+ bφ u2 hx, Dφi dx
=−
BR 2
Z
Z
n
R
2
−
(a + bφ)φu dx +
(a + bφ)φu2 dσ;
2 BR
2 ∂BR
Z
Z
Z
g(u)hx, Dui dx = −n
G(u) dx + R
G(u)dσ;
BR
BR
∂BR
Z
Z
∆2k φ hx, Dφi dx =
D(|Dφ|2k−2 Dφ)hx, Dφi dx
BR
BR
Z
Z
R
n − 2k
|Dφ|2k dx −
|Dφ|2k dσ
=
2k
2k
BR
∂BR
Z
1
2k−2
2
|Dφ|
hx, Dφi dσ,
+
R ∂BR
BR
(3.2)
(3.3)
(3.4)
where ∆2k φ = D(|Dφ|R2k−2 |Dφ|) and g : R → R is a continuous function such that
s
g(0) = 0 and G(s) = 0 g(t) dt.
Proof. The proofs of (3.1), (3.2) and (3.3) can be found in [9, Lemma 3.1]. In
the following we show (3.4). For fix i1 , . . . , ik−1 , j, l = 1, 2, . . . , n, we see from the
integration by parts formula that
Z
φ2xi1 . . . φ2xi φxj xj xl φxl dx
k−1
BR
Z
Z
xj
=−
(φ2xi1 . . . φ2xi xl φxl )xj φxj dx +
φ2xi1 . . . φ2xi xl φxl φxj dσ
k−1
k−1
|x|
B
∂B
Z R
Z R
(φ2xi1 . . . φ2xi )xj xl φxl φxj dx −
φ2xi1 . . . φ2xi φxl φxj δlj dx
=−
k−1
k−1
BR
BR
Z
Z
xj
−
φ2xi1 . . . φ2xi xl φxl xj φxj dx +
φ2xi1 . . . φ2xi xl φxl φxj dσ,
k−1
k−1
|x|
BR
∂BR
where dσ indicates the (n − 1)-dimensional area element in ∂BR and δlj are the
Kroneker symbols. Summing up for i1 , . . . , ik−1 , j, l = 1, 2, . . . , n, we have
Z
|Dφ|2k−2 ∆φhx, Dφi dx
BR
Z
Z
=−
hD|Dφ|2k−2 , Dφihx, Dφi dx −
|Dφ|2k dx
(3.5)
B
B
Z R
Z R
1
|Dφ|2k−2 hx, D2 φDφi dx +
|Dφ|2k−2 hx, Dφi2 dσ.
−
R
BR
∂BR
Similarly, for fix i1 , . . . , ik−1 , j, l = 1, 2, . . . , n, we see from the integration by parts
formula that
Z
Z
2
φ2xi1 . . . φ2xi xl φxl xj φxj dx =
φ2xi1 . . . φ2xi xl (φ2xj )xl dx
k−1
BR
Z
=−
BR
BR
(φ2xi1 . . . φ2xi
k−1
xl )xl φ2xj dx +
Z
∂BR
k−1
φ2xi1 . . . φ2xi
k−1
φ2xj
x2l
dσ
|x|
EJDE-2012/82
SOLITARY WAVES
Z
=−
BR
(φ2xi1 . . . φ2xi
Z
+
∂BR
k−1
φ2xi1 . . . φ2xi
)xl xl φ2xj dx −
k−1
φ2xj
Z
BR
7
(φ2xi1 . . . φ2xi
k−1
)xl φ2xj dx
x2l
dσ.
|x|
Summing up for i1 , . . . , ik−1 , j, l = 1, 2, . . . , n, we have
Z
|Dφ|2k−2 hx, D2 φDφidx
BR
Z
Z
Z
=−
hx, D(|Dφ|2k−2 )i|Dφ|2 dx − n
|Dφ|2k dx + R
|Dφ|2k dσ
BR
BR
∂BR
Z
Z
Z
2k−2
2
2k
= −2(k − 1)
|Dφ|
hx, D φDφidx − n
|Dφ| dx + R
|Dφ|2k dσ.
2
BR
BR
∂BR
Then
Z
|Dφ|2k−2 hx, D2 φDφidx = −
BR
n
2k
Z
BR
|Dφ|2k dx +
R
2k
Z
|Dφ|2k dσ.
(3.6)
∂BR
Using (3.5) and (3.6), we obtain
Z
∆2k φ hx, Dφi dx
BR
Z
D(|Dφ|2k−2 Dφ)hx, Dφi dx
B
Z R
Z
=
|Dφ|2k−2 ∆φ hx, Dφi dx +
hD|Dφ|2k−2 , Dφihx, Dφi dx
BR
BR
Z
Z
Z
1
2k
2k−2
=−
|Dφ| dx −
|Dφ|
hx, D2 φDφi dx +
|Dφ|2k−2 hx, Dφi2 dσ
R
BR
BR
∂BR
Z
Z
Z
n − 2k
R
1
2k
2k
=
|Dφ| dx −
|Dφ| dσ +
|Dφ|2k−2 hx, Dφi2 dσ.
2k
2k ∂BR
R ∂BR
BR
=
Set Ω = m2 − w2 . From the above Lemma we have the following result.
Lemma 3.2. If (u, φ) is a solution of the system (1.6), then (u, φ) satisfies the
Pohožaev type identity:
Z
N
1 X 3(k − 1)
βk
|Dφ|2k dx
4π
k
R3
R3
R3
k=2
Z
Z
Z
6
−2
φ2 u2 dx + 5
ωφu2 dx −
|u|q dx = 0.
q R3
3
3
R
R
Z
|Du|2 dx + 3
Z
u2 dx +
(3.7)
8
F. WANG
EJDE-2012/82
Proof. Multiplying the first formula of (1.6) by hx, Dui, integrating on BR and
using the above Lemma, we conclude that
Z
Z
1
3
−
|Du|2 dx − Ω
u2 dx
2 BR
2
BR
Z
Z
Z
3
3
(φ − 2ω)φu2 dx +
|u|q dx
+
(φ − ω)u2 hx, Dφi dx +
2 BR
q BR
BR
Z
Z
(3.8)
R
1
hx, Dui2 dσ −
|Du|2 dσ
=
R ∂BR
2 ∂BR
Z
Z
Z
ΩR
R
R
2
2
−
u dσ +
(φ − 2ω)φu dσ +
|u|q dx.
2 ∂BR
2 BR
q ∂BR
Multiplying the second formula of (1.6) by hx, Dφi, integrating on BR and using
the above Lemma, we obtain
Z
(φ − ω)u2 hx, Dφi dx
4π
BR
N
X
Z
=
(βk ∆2k φ)hx, Dφi dx
BR k=1
=
N
X
Z
βk
=
N
X
(3.9)
∆2k φhx, Dφi dx
BR
k=1
3 − 2k Z
βk
R
2k
2k
BR
|Dφ|2k−2 hx, Dφi2 dσ .
k=1
1
+
R
Z
|Dφ|2k dx −
Z
|Dφ|2k dσ
∂BR
∂BR
By (3.8), (3.9) and the proof of [9, Theorem 1.1, pp. 316-317], we deduce the
equality
Z
Z
N
3
1 X 3 − 2k
2
|Du| dx − Ω
u dx +
|Dφ|2k dx
βk
2
4π
2k
R3
R3
R3
k=1
Z
Z
3
3
+
(φ − 2ω)φu2 dx +
|u|q dx = 0.
2 R3
q R3
1
−
2
Z
2
Then, noting (1.6), we have
Z
N
1 X 3(k − 1)
|Dφ|2k dx
βk
2π
2k
R3
R3
R3
k=2
Z
Z
Z
6
φ2 u2 dx + 5ω
φu2 dx −
−2
|u|q dx = 0.
q R3
3
3
R
R
Z
|Du|2 dx + 3Ω
Z
u2 dx +
4. Proof of the main theorem
First, we give a abstract result which is due to Jeanjean [13].
EJDE-2012/82
SOLITARY WAVES
9
Proposition 4.1. Let (X, k · k) be a Banach space and let I ⊂ R+ be an interval.
Consider the family of C 1 functionals on X
Ψλ (u) = A(u) − λB(u),
∀λ ∈ I,
with B(u) nonnegative and either A(u) → +∞ or B(u) → +∞, as kuk → ∞ and
such that Ψλ (0) = 0. For any λ ∈ I we set
Γλ = {γ ∈ C([0, 1], X) : γ(0) = 0, Ψλ (γ(1)) ≤ 0}.
If for every λ ∈ I the set Γλ is nonempty and
cλ := inf max Ψλ (γ(t)) > 0,
γ∈Γλ t∈[0,1]
then for almost every λ ∈ I there is a sequence {(uλ )n } ⊂ X such that
(i) {(uλ )n } is bounded in X;
(ii) Ψλ ((uλ )n ) → cλ ;
(iii) Ψ0λ ((uλ )n ) → 0 in the dual X ∗ of X.
Proof Theorem 1.1. Denote
M (φ) :=
Z
N
1 X k −1
|Dφ|2k dx.
βk
4π
k
R3
k=2
Then, noting the definition of Φ(u) we can write (3.7) and J(u) by:
Z
Z
Z
|Du|2 dx + 3Ω
u2 dx + 3M (Φ(u)) − 2
Φ2 (u)u2 dx
3
3
3
R
R
R
Z
Z
6
2
q
+ 5ω
Φ(u)u dx −
|u| dx = 0
q R3
R3
and
JN (u) =
Z
Z
1
ω
|Du|2 dx + Ω
u2 dx +
Φ(u)u2 dx
2
2
3
3
3
R
R
Z R
1
1
q
+ M (Φ(u)) −
|u| dx,
2
q R3
1
2
Z
respectively.
For λ ∈ [ 21 , 1], we define the family of functionals JN,λ : Hr1 (R3 ) → R by
Z
Z
Z
1
ω
1
|Du|2 dx + Ω
u2 dx +
Φ(u)u2 dx
JN,λ (u) =
2 R3
2
2 R3
R3
Z
1
λ
+ M (Φ(u)) −
|u|q dx
2
q R3
Using a slightly modified version of [2, Lemmas 2.3 and 2.4], it can be proved that:
for every λ ∈ [ 21 , 1], there exist αλ , ρλ > 0 and νλ ∈ Hr1 (R3 ) such that
(i) inf kuk=ρλ JN,λ (u) > αλ .
(ii) kνλ k > ρλ and JN,λ (νλ ) < 0.
Thus JN,λ has the mountain pass geometry. So we can define the Mountain Pass
level cλ by
cλ := inf max JN,λ (γ(t)),
γ∈Γλ 0≤t≤1
where
Γλ = {γ ∈ C([0, 1], Hr1 (R3 )) : γ(0) = 0, γ(1) = νλ }.
10
F. WANG
EJDE-2012/82
Set X = Hr1 (R3 ), I = [ 12 , 1], Ψλ = JN,λ ,
Z
Z
Z
1
1
ω
1
2
2
A(u) =
|Du| dx + Ω
u dx +
Φ(u)u2 dx + M (Φ(u))
2 R3
2
2
2
3
3
R
R
and
Z
1
B(u) =
|u|q dx.
q R3
It is easy to see that B(u) ≥ 0 for all u ∈ Hr1 (R3 ) and A(u) → +∞ as kuk → ∞.
Thus, by Proposition 4.1, for almost every λ ∈ I there is a sequence {(uλ )n } ⊂ X
such that
(i) {(uλ )n } is bounded in Hr1 (R3 );
(ii) JN,λ ((uλ )n ) → cλ ;
0
(iii) JN,λ
((uλ )n ) → 0 in the dual (Hr1 (R3 ))∗ of Hr1 (R3 ).
There exists uλ ∈ Hr1 (R3 ) such that
Jλ0 (uλ ) = 0,
Jλ (uλ ) = cλ ,
for almost every λ ∈ I. Now we can choose a suitable λn → 1 and uλn such that
Jλ0 n (uλn ) = 0,
Jλn (uλn ) = cλn → c1 ,
For simplicity we denoted uλn by un . Since Jλ0 n (un ) = 0, un satisfies the Pohožaev
equality
Z
Z
Z
|Dun |2 dx + 3Ω
u2n dx + 3M (Φ(un )) − 2
Φ2 (un )u2n dx
R3
R3
R3
Z
Z
(4.1)
6λn
2
+ 5ω
Φ(un )un dx −
|un |q dx = 0.
q R3
R3
By Jλ0 n (un ) = 0 and Jλn (un ) = cλn → c1 , we have
Z
Z
Z
|∇un |2 dx + Ω
u2n dx + 2ω
Φ(un )u2n dx
R3
R3
R3
Z
Z
2
2
−
Φ (un )un dx − λn
|un |q dx = 0
R3
(4.2)
R3
and, for n large enough,
Z
Z
1
1
1
|∇un |2 dx + Ω
u2n dx + M (Φ(un ))
2 R3
2
2
3
R
Z
Z
ω
λn
2
+
Φ(un )un dx −
|un |q dx ≤ c1 + 1.
2 R3
q R3
Set α and β two real number (which we will estimate later). Then from α × (4.1) +
β × (4.2), we obtain
Z
λn
|un |q dx
q R3
Z
Z
n
1
2
=
(α + β)
|∇un | dx + (3α + β)Ω
u2n dx + 3αM (Φ(un ))
6α + qβ
R3
R3
Z
Z
o
+ (5α + 2β)
ωΦun u2n dx − (2α + β)
Φ2un u2n dx .
R3
Thus
c1 + 1 ≥ Jλn (un )
R3
EJDE-2012/82
SOLITARY WAVES
11
Z
Z
1
|∇un |2 dx + Ω
u2n dx
2
R3
R3
Z
Z
1
1
λn
2
+ M (Φ(un )) +
ωφun un dx −
|un |q dx
2
2 R3
q R3
Z
Z
1
1
α+β 3α + β =
|∇un |2 dx +
−
−
Ω
u2n dx
2 6α + qβ R3
2 6α + qβ
3
R
1 5α + 2β Z
1
3α 2
M (Φ(un ))
+
−
ωφun un dx +
−
2 6α + qβ
2 6α + qβ
R3
Z
2α + β
Φ2 (un )u2n dx
+
6α + qβ R3
Z
1
τ +1 =
−
|∇un |2 dx
2 6τ + q R3
Z
1
2τ + 1
3τ +
M (Φ(un )) +
−
Φ2 (un )u2n dx
2 6τ + q
6τ + q R3
1 5τ + 2 Z
1 3τ + 1 Z
2
−
Ω
un dx +
−
ωΦ(un )u2n dx,
+
2 6τ + q
2 6τ + q R3
R3
where τ = α . Under one of the following conditions:
β
(i) q ∈ (4, 6), τ ∈ ((2 − q)/4, −1/2) and m > ω > 0;
(ii) q ∈ (3, 4], τ ∈ ((2 − q)/4, (q − 4)/4) and
p m > ω > 0;
(iii) q ∈ (2, 3], τ ∈ ((2 − q)/4, +∞) and m (q − 2)(4 − q) > ω > 0,
=
1
2
we conclude that
1
τ +1
−
> 0,
2 6τ + q
1
3τ
−
>0
2 6τ + q
and
1 3τ + 1 2τ + 1 2 1 5τ + 2 t +
−
ωt +
−
Ω ≥ 0, for t ∈ [0, ω].
6τ + q
2 6τ + q
2 6τ + q
R
So we obtain that R3 |∇un |2 dx is bounded for all n. Then, as in [2, Proof of
Teorem 1.1, pp. 9] we have {un } is bounded in Hr1 (R3 ). Thus {un } is a bounded
(P S)c1 -sequence for JN . So JN has a nontrivial critical point uN .
References
[1] R. A. Adams; Sobolev spaces, Academic Press, New York, 1975.
[2] A. Azzollini, L. Pisani, A. Pomponio; Improved estimates and a limit case for the electrostatic
Klein-Gordon-Maxwell system, Proceedings of the Royal Society of Edinburgh: Section A
Mathematics 141(2011), 449–463.
[3] A. Azzollini, A. Pomponio; Ground state solutions for the nonlinear Klein-Gordon-Maxwell
equations, Topol. Methods Nonlinear Anal. 35(2010) 33–42.
[4] V. Benci, D. Fortunato; Solitary waves of the nonlinear Klein-Gordon equation coupled wiht
the Maxwell equations, Rev. Math. Phys. 14(2002), 409–420.
[5] Benci, Fortunato; Existence of hylomorphic solitary waves in Klein-Gordon and in KleinGordon-Maxwell equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei
(9) Mat. Appl. 20 (2009) 243–279.
[6] Benci, Fortunato; Three-dimensional vortices in abelian gauge theories, Nonlinear Anal. 70
(2009) 4402–4421.
[7] Benci, Fortunato; Spinning Q-balls for the Klein-Gordon-Maxwell equations, Comm. Math.
Phys. 295 (2010) 639–668.
[8] M. Born, L. Infeld; Foundation of the new field theory, Proc. Roy. Soc. A 144(1934) 425–451.
12
F. WANG
EJDE-2012/82
[9] T. d’Aprile, D. Mugnai; Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud. 4(2004) 307–322.
[10] T. d’Aprile, D. Mugnai; Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger
-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 893–906.
[11] P. d’Avenia, L. Pisani; Nonlinear Klein-Gordon equations coupled with Born-Infeld type
equations, Elect. J. Diff. Eqns. 26 (2002) 1–13.
[12] D. Fortunato, L. Orsina, L. Pisani; Born-Infeld type equations for electrostatic fields, J. of
Math. Phys. 43(11)(2002) 5698–5706.
[13] L. Jeanjean; On the existence of bounded Palais-Smale sequence and application to a
Landesman-Lazer type problem set on RN , Proc. Roy. Soc. Edinburgh Sect. A 129 (1999)
787–809.
[14] D. Mugnai; Coupled Klein-Gordon and Born-Infeld type equations: Looking for solitary
waves, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004) 1519–1528.
[15] Feizhi Wang; Ground-state solutions for the electrostatic nonlinear Klein-Gordon-Maxwell
system, Nonlinear Analysis 74(2011) 4796–4803.
[16] R. S. Palais; The principle of symmetric criticality, Comm. Math, Phys. 69(1979) 19–30.
[17] M. Willem; Minimax Theorems. Birkhäuser, Boston, 1996.
[18] Y. Yu; Solitary waves for nonlinear Klein-Gordon equations coupled with Born-Infeld theory,
Ann. I. H. Poincaré–AN 27 (2010) 351–376.
Feizhi Wang
School of Mathematics, Yantai University, Yantai, Shandong, China
E-mail address: wangfz@ytu.edu.cn