PORTUGALIAE MATHEMATICA
Vol. 63 Fasc. 1 – 2006
Nova Série
SOLUTIONS FOR SINGULAR CRITICAL GROWTH
SCHRÖDINGER EQUATIONS WITH MAGNETIC FIELD
Pigong Han
Recommended by Luı́s Sanchez
Abstract: In this paper, we consider the semilinear stationary Schrödinger equation
∗
with a magnetic field: −∆A u − V (x)u = |u|2 −2 u in RN , where A is the vector (or
magnetic) potential and V is the scalar (or electric) potential. By means of variational
method, we establish the existence of nontrivial solutions in the critical case.
1 – Introduction and main result
In this paper, we are concerned with the semilinear Schrödinger equation
(1.1)
∗ −2
−∆A u − V (x)u = |u|2
u,
x ∈ RN ,
where −∆A = (−i∇+A)2 , u : RN → C, N ≥ 3, 2∗ = N2N
−2 denotes the critical
Sobolev exponent, A = (A1 , A2 , ..., AN ) : RN → RN is the vector (or magnetic)
potential, the coefficient V is the scalar (or electric) potential and may be signchanging.
The nonlinear Schrödinger equation arises in different physical theories (e.g.,
the description of Bose–Estein condensates and nonlinear optics), and has been
widely considered in the literature, see [1, 6, 7, 8, 11, 13].
Throughout this paper, suppose A ∈ L2loc (RN , RN ). Define
Z
2
N
−
N
2 −
L (R , V dx) := u : R → C
|u| V dx < ∞
RN
and
2
N
1
N
2
N
−
HA,V − (R ) := u ∈ L (R , V dx) ∇A u ∈ L (R ) ,
where ∇A = (∇ + iA), V ± = max{±V, 0} 6≡ 0.
Received : November 19, 2004; Revised : December 29, 2004.
Keywords: Schrödinger equation; energy functional; (P.S.) sequence; critical Sobolev
exponent.
38
PIGONG HAN
1
N
HA,V
− (R ) is a Hilbert space with the inner product
Z
∇A u · ∇A v + V − uv dx ,
RN
where the bar denotes complex conjugation.
N
1
It is known that C0∞ (RN ) is dense in HA,V
− (R ) (see [8]).
1
N
Definition 1.1. u ∈ HA,V
− (R ) is said to be a weak solution of problem
(1.1) if
Z ∗
1
N
(1.2)
∇A u · ∇A ϕ − V (x) u ϕ − |u|2 −2 u ϕ dx = 0 ∀ ϕ ∈ HA,V
− (R ) .
RN
The corresponding energy functional of problem (1.1) is defined by
Z
Z
1
1
∗
1
N
2
2
|u|2 dx, u ∈ HA,V
(1.3) IA,V (u) =
|∇A u| − V (x)|u| dx − ∗
− (R ) .
2 RN
2 RN
It is well known that the nontrivial solutions of problem (1.1) are equivalent
N
1
to the nonzero critical points of IA,V in HA,V
− (R ).
Now we list some assumptions on the potential V :
N
N
(A1 ) 0 6≡ V − ∈ L 2 (RN ) and V ∈ L 2 (RN \BR (0)) for any R > 0. Moreover,
there exist δ > 0, λ > 0 such that
|x|2 V (x) = µ + λ|x|α ,
∀ x ∈ Bδ (0) ,
√
where 0 < µ < µ = ( N 2−2 )2 and 0 < α < min{2, 2 µ − µ}.
(A2 ) There is θ ∈ (0, 1) such that
Z
Z
+
2
V (x) |u| dx ≤ θ
|∇A u|2 + V − |u|2 dx
RN
RN
1
N
for any u ∈ HA,V
− (R ) .
Remark. There does exist such potential V satisfying assumptions (A1 ), (A2 ).
For example, take 0 < δ < 1
 µ
α−2

if |x| ≤ δ ,

 |x|2 + λ|x|
V (x) =
k


if |x| > δ ,
− β
|x|
where 0 < λ < µ − µ, β > 2 and k > 0.
SCHRÖDINGER EQUATIONS WITH MAGNETIC FIELD
39
A direct computation shows that V (x) satisfies assumptions (A1 ), (A2 ).
Our main result is the following:
Theorem 1.1. Assume that (A1 ), (A2 ) hold, and A is continuous at 0.
Then problem (1.1) admits at least one nontrivial solution.
We prove Theorem 1.1 by critical point theory. However, since the functional
IA,V does not satisfy the Palais–Smale condition due to the lack of compactness
N
2∗ (RN ), the standard variational argument is
1
of the embedding: HA,V
− (R ) ֒→ L
not applicable directly. In addition, from assumption (A1 ), the potential V has a
strong singularity at the origin, which also brings some difficulty in dealing with
1
2
−2 dx) is continuous but
(1.1). Precisely, the embedding: HA,V
− (Ω) ֒→ L (Ω, |x|
not compact, where Ω ∋ 0 is an arbitrary bounded set in RN . Nevertheless, we
can prove that IA,V satisfies the (P.S.)c condition with c below some energy level.
We need to construct a suitable (P.S.)c compact sequence, which is obtained by
the mountain-pass theorem (see [3]).
1
N
Throughout this paper, we shall denote the norm of the space HA,V
− (R ) by
R
1
kukH 1 − (RN ) = ( RN (|∇A u|2 +V − |u|2 ) dx) 2 , and the positive constants (possibly
A,V
different) by C, C1 , C2 , ... .
2 – Proof of Theorem 1.1
Before giving the proof of Theorem 1.1, we introduce some notations and
preliminary lemmas.
Set
Z u2
2
|∇u| − µ 2 dx
|x|
RN
Sµ :=
inf
.
2
Z
u∈D1,2 (RN )\{0}
∗
RN
|u|2 dx
2∗
From [9, 10], Sµ is independent of any Ω ⊂ RN in the sense that if
Z u2
2
|∇u| − µ 2 dx
|x|
Ω
,
Sµ (Ω) :=
inf
2
Z
u∈H01 (Ω)\{0}
∗
Ω
then Sµ (Ω) = Sµ (RN ) = Sµ .
|u|2 dx
2∗
40
PIGONG HAN
√
√
√
√
Let µ̄ = ( N 2−2 )2 , γ = µ̄ + µ̄ − µ, γ ′ = µ̄ − µ̄ − µ, F. Catrina and
Z.Q. Wang [5], S. Terracini [14] proved that for ǫ > 0
Uǫ (x) =
satisfies
(
N −2
4
4 ǫ2 N (µ̄ − µ)/(N − 2)
√µ̄
γ′
√
√γ
ǫ2 |x| µ̄ + |x| µ̄
∗ −2
−∆u = |u|2
u + µ |x|u2
in RN \{0} ,
u→0
as |x| → ∞ .
Moreover, Uǫ achieves Sµ .
Lemma 2.1. For any bounded set Ω ⊂ RN , 0 ∈ Ω, the embedding: H 1 (Ω) ֒→
L2 (Ω, |x|l ) is compact with l > −2.
Proof: Let {um } ⊂ H 1 (Ω) be a bounded sequence. Then, up to a subsequence, we may assume
um ⇀ u
um → u
weakly in H 1 (Ω) ;
strongly in Lp (Ω) with 1 < p < 2∗ ;
um → u
Choose max{2,
Z
2N
N +l }
2
l
Ω
< q < 2∗ . Then
|x| |um − u| dx ≤
Z
Ω
|x|
By the choice of q, we easily have
Z
lq
|x| q−2 dx ≤ C and
Ω
Thus, lim
R
|x|
m→∞ Ω
l |u
m
a.e. in Ω .
lq
q−2
q−2 Z
q
dx
lim
Z
m→∞ Ω
q
Ω
|um − u| dx
2
q
.
|um − u|q dx = 0 .
− u|2 dx = 0.
Lemma 2.2. The functional IA,V satisfies the (P.S.)c condition with
c<
1
N
N
Sµ2 .
SCHRÖDINGER EQUATIONS WITH MAGNETIC FIELD
41
1
N
Proof: Assume that {um } ⊂ HA,V
− (R ) satisfies
IA,V (um ) → c
and
dIA,V (um ) → 0 as m → ∞ .
Then, by assumption (A2 ), we get
Z 1
1
|∇A um |2 + V − |um |2 dx =
− ∗
2 2
RN
Z
1
1
1
= IA,V (um ) − ∗ dIA,V (um ), um +
V + (x) |um |2 dx
−
2
2 2∗
N
R
Z 1
1
−
θ
|∇A um |2 + V − |um |2 dx + o(1) ,
≤ c+
2 2∗
RN
Z
which implies
|∇A um |2 + V − |um |2 dx ≤ C.
RN
By choosing a subsequence if necessary, we may assume that
N
1
weakly in HA,V
− (R )
um ⇀ u
and
um → u
a.e. on RN .
It is easy to verify that u is a weak solution of problem (1.1). Set um = vm +u.
Then
Z |∇A vm |2 − V |vm |2 dx =
RN
Z Z =
|∇A um |2 − V |um |2 dx −
|∇A u|2 − V |u|2 dx + o(1)
RN
RN
and by Brezis–Lieb lemma (see [2])
Z
Z
Z
2∗
2∗
|um | dx −
|vm | dx =
RN
RN
RN
∗
|u|2 dx + o(1) .
Therefore, we get
dIA,V (vm ), vm = dIA,V (um ), um − dIA,V (u), u + o(1) = o(1) .
Thus,
(2.1)
lim
Z
m→∞ RN
2
|∇A vm | − V |vm |
2
dx = lim
m→∞ RN
where a is a nonnegative number.
If a = 0, then we infer
Z Z
2
−
2
lim
|∇A vm | + V |vm | dx = lim
m→∞ RN
Z
∗
|vm |2 dx = a ,
V + |vm |2 dx
R
ZN 2
−
2
≤ θ lim
|∇A vm | + V |vm | dx ,
m→∞
m→∞
RN
42
PIGONG HAN
which implies
(2.2)
lim
Z
m→∞ RN
|∇A vm |2 + V − |vm |2 dx = 0 .
If a > 0 then, by Sobolev inequality, we obtain
Z
2∗
Z
2
−1
|vm | dx ≤ Sµ
µ|vm |2
dx
∇|vm | −
|x|2
RN
Z µ|vm |2
−1
2
≤ Sµ
|∇A vm | −
dx
|x|2
RN
Z
Z
µ|vm |2
2
−1
|∇A vm | dx −
≤ Sµ
dx
2
Bδ (0) |x|
RN
Z −1
≤ Sµ
|∇A vm |2 − V (x) |vm |2 dx
RN Z
Z
−1
α−2
2
−1
+ λ Sµ
|x|
|vm | dx + Sµ
2∗
RN
(2.3)
2
RN \B
Bδ (0)
V (x) |vm |2 dx ,
δ (0)
where we use the diamagnetic inequality in the above argument (see [12]):
∇|u| ≤ |∇A u|
a.e. in RN .
2
Hence, by assumption (A1 ), Lemma 2.1 and (2.1), (2.3), we derive a 2∗ ≤ Sµ−1 a,
N
and then a ≥ Sµ2 .
In addition,
IA,V (u) = IA,V (u) −
1
1
dIA,V (u), u =
2
N
Z
RN
∗
|u|2 dx ≥ 0 .
Therefore,
c = IA,V (um ) + o(1)
= IA,V (vm ) + IA,V (u) + o(1)
Z Z
1
1
∗
|vm |2 dx + o(1)
≥
|∇A vm |2 − V (x) |vm |2 dx − ∗
2 RN
2 RN
1
1
=
− ∗ a
2 2
1 N2
≥
Sµ ,
N
which contradicts c <
1
N
N
Sµ2 .
43
SCHRÖDINGER EQUATIONS WITH MAGNETIC FIELD
Define
cA = inf max IA,V (γ(t)) ,
γ∈Γ t∈[0,1]
1
N
where Γ = γ ∈ C [0, 1], HA,V
− (R )
γ(0) = 0, IA,V (γ(1)) < 0 .
Lemma 2.3. Let the assumptions of Theorem 1.1 hold. Then cA <
1
N
N
Sµ2 .
Proof: Since A(x) is continuous at 0, we infer |A(x)| ≤ c0 for all |x| ≤ η(≤ δ).
Set uǫ (x) = ψ(x) Uǫ (x), where ψ is a cut off function satisfying ψ(x) ≡ 1 if |x| ≤ η2 ,
ψ(x) ≡ 0 if |x| ≥ η and 0 ≤ ψ(x) ≤ 1.
Following [3] and after a detailed calculation, we have the following estimates:
N
|ψUǫ |2
2
2
|∇(ψUǫ )| − µ
dx
=
S
+ O(ǫN −2 ) ,
µ
|x|2
RN
Z
N
∗
|ψUǫ |2 dx = Sµ2 + O(ǫN ) ,
Z
(2.4)
(2.5)
RN
(2.6)
(2.7)
 √N −2
µ−µ ,

if 0 < µ < µ − 1 ,

ǫ
2
N
−2
|ψUǫ | dx ≈ β(ǫ) =
ǫ
| log ǫ| if µ = µ − 1 ,

RN

 N −2
ǫ
if µ − 1 < µ < µ ,
Z
√
α µ
√
|x|α−2 |ψUǫ |2 dx ≈ ǫ µ−µ ,
Z
RN
where Aǫ ≈ Bǫ means C1 Bǫ ≤ Aǫ ≤ C2 Bǫ .
Observe that
Z |∇A uǫ |2 − V (x) |uǫ |2 dx =
RN
Z =
|∇(ψUǫ )|2 + |A|2 |ψUǫ |2 − V (x) |ψUǫ |2 dx
RN
(2.8)
|ψUǫ |2
2
=
|∇(ψUǫ )| − µ
dx
|x|2
RN
Z
Z
|x|α−2 |ψUǫ |2 dx
|A|2 |ψUǫ |2 dx − λ
+
Z
RN
RN
N
≤ Sµ2 − C1 ǫ
where β(ǫ) is given by (2.6).
√
α µ
√
µ−µ
+ C2 β(ǫ) + O(ǫN −2 ) ,
44
PIGONG HAN
Therefore, from (2.4)–(2.8), we conclude
cA ≤ max IA,V (tuǫ )
t≥0
!N
|∇A uǫ |2 − V (x) |uǫ |2 dx 2
2∗
R
2∗
2
RN |uǫ | dx
1
=
N
R
1
≤
N
Sµ2 − C1 ǫ
RN
N
<
√
α µ
√
µ−µ
+ C2 β(ǫ) + O(ǫN −2 )
N −2
Sµ 2 + O(ǫN −2 )
1 N2
Sµ
N
!N
2
p
by the choice of α : 0 < α < min 2, 2 µ − µ .
N
1
Proof of Theorem 1.1: By assumption (A2 ), for any u ∈ HA,V
− (R ), we
have
Z Z
1
1
∗
2
2
IA,V (u) =
|u|2 dx
|∇A u| − V (x) |u| dx − ∗
2 RN
2 RN
∗
Z Z 22
1−θ
2
−
2
2
−
2
|∇A u| + V |u| dx − C
|∇A u| + V |u| dx .
≥
2
RN
RN
Thus, there exists a sufficiently small constant ρ > 0 such that
b(u) :=
kukH 1
inf
(R
A,V −
N ) =ρ
IA,V (u) > 0 = IA,V (0) .
1
N
In addition, for any v ∈ HA,V
− (R )\{0}, IA,V (tv) → −∞ as t → ∞. Hence,
there is a t0 > 0 such that kt0 vk > ρ and IA,V (t0 v) < 0. By using a variant of
1
N
the mountain pass theorem (see [3]), there exists a sequence {um } ⊂ HA,V
− (R )
such that as m → ∞
IA,V (um ) → cA ,
dIA,V (um ) → 0 .
N
1
By Lemmas 2.2, 2.3, the sequence {um } is relatively compact in HA,V
− (R ).
1
N
So there exist a subsequence, still denoted by {um }, and a function u ∈ HA,V
− (R )
such that
1
N
um → u strongly in HA,V
− (R ) .
Thus cA is a critical value of IA,V , and u is a corresponding critical point of IA,V
1
N
in HA,V
− (R ).
SCHRÖDINGER EQUATIONS WITH MAGNETIC FIELD
45
ACKNOWLEDGEMENTS – The author would like to thank the anonymous referee of
this paper for very helpful comments.
REFERENCES
[1] Arioli, G. and Szulkin, A. – A semilinear Schrödinger equation in the presence
of a magnetic field, Arch. Rational Mech. Anal., 170 (2003), 277–295.
[2] Brezis, H. and Lieb, E. – Relation between pointwise convergence of functions
and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486–490.
[3] Brezis, H. and Nirenberg, L. – Positive solutions of nonlinear elliptic equations
involving critical Sobolev exponent, Comm. Pure Appl. Math., 36 (1983), 437–478.
[4] Byeon, J. and Wang, Z.Q. – Standing waves with a critical frequency for nonlinear Schrödinger equation, Arch. Rational Mech. Anal., 165 (2002), 295–316.
[5] Catrina, F. and Wang, Z.Q. – On the Caffarelli–Kohn–Nirenberg inequalities:
sharp constants, existence (and nonexistence), and symmetry of extermal functions,
Comm. Pure Appl. Math., 54 (2001), 229–258.
[6] Chabrowski, J. andSzulkin, A. – On the Schrödinger equation involving a
critical Sobolev exponent and magnetic field, Topol. Methods Nonlinear Anal., 25
(2005), 3–21.
[7] Cingolani, S. and Secchi, S. – Semiclassical limit for nonlinear Schrödinger
equations with electromagnetic fields, J. Math. Anal. Appl., 275 (2002), 108–130.
[8] Esteban, M.J. and Lions, P.L. – Stationary solutions of nonlinear Schrödinger
equations with an external magnetic field , In: “Partial Differential Equations and
the Calculus of Variations” (F. Colombini, A. Marino, L. Modica and S. Spagnolo,
Eds.), Vol. 1, Birkhäuser, 1989, pp. 401–449.
[9] Ferrero, A. and Gazzola, F. – Existence of solutions for singular critical growth
semilinear elliptic equations, J. Differential Equations., 177 (2001), 494–522.
[10] Jannelli, E. – The role played by space dimension in elliptic critical problems,
J. Differential Equations, 156 (1999), 407–426.
[11] Kurata, K. – Existence and semiclassical limit of the least energy solution to a
nonlinear Schrödinger equations with electromagnetic fields, Nonlinear Anal., 41
(2000), 763–778.
[12] Lieb, E.H. and Loss, M. – Analysis, Graduate Studies in Mathematics 14, AMS
(1997).
[13] Schindler, I. and Tintarev, K. – A nonlinear Schrödinger equations with
external magnetic field, Rostock Math. Kolloq., 56 (2002), 49–54.
[14] Terracini, S. – On positive solutions to a class equations with a singular coefficient and critical exponent, Adv. Differential Equations., 2 (1996), 241–264.
Pigong Han,
Institute of Applied Mathematics, Academy of Mathematics and Systems Science,
Chinese Academy of Sciences, Beijing 100080 – PEOPLE’S REPUBLIC OF CHINA
E-mail: pghan@amss.ac.cn