Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 75, pp. 1–15.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
SEMI-CLASSICAL STATES FOR SCHRÖDINGER-POISSON
SYSTEMS ON R3
HONGBO ZHU
Abstract. In this article, we study the nonlinear Schrödinger-Poisson equation
−2 ∆u + V (x)u + φ(x)u = f (u),
−2 ∆φ = u2 ,
x ∈ R3 ,
lim φ(x) = 0 .
|x|→∞
Under suitable assumptions on V (x) and f (s), we prove the existence of ground
state solution around local minima of the potential V (x) as → 0. Also, we
show the exponential decay of ground state solution.
1. Introduction
Consider the nonlinear Schrödinger equation:
∂ψ
i
= −2 ∆ψ + Ṽ ψ − f (ψ)
∂t
coupled with the Poisson equation
− 2 ∆φ = |ψ|2 ,
(1.1)
(1.2)
where is the planck constant, i is the imaginary unit and Ṽ , ψ are real functions on R3 and represent the effective potential and electric potential respectively.
ψ(x, t) → C and f is supposed to satisfy f (αeiθ ) = f (α)eiθ for all θ, α ∈ R. Problem
(1.1), (1.2) arose from semiconductor theory; see e.g, [4, 10, 25] and the references
therein for more physical background.
We are interested in standing wave solutions, namely solutions of form ψ(x, t) =
u(x) exp(iωt/) with u(x) > 0 in R3 and ω > 0 (the frequency), then it is not
difficult to see that u(x) must satisfy
−2 ∆u + V (x)u + φ(x)u = f (u),
x ∈ R3 ,
−2 ∆φ = u2 , lim φ(x) = 0.
(1.3)
|x|→∞
An interesting class of solutions of (1.3), sometimes called semi-classical states,
are families solutions u (x) which concentrate and develop a spike shape around
one, or more, special points in R3 , which vanishing elsewhere as → 0.
2010 Mathematics Subject Classification. 35B38, 35J20, 35J50.
Key words and phrases. Schrödinger-Poisson system; semi-classical states; variational method.
c
2016
Texas State University.
Submitted August 14, 2015. Published March 17, 2016.
1
2
H. ZHU
EJDE-2016/75
Similar equations have been studied extensively by many authors concerning
existence, non-existence, multiplicity when the nonlinearity f (u) = up , 1 < p < 5
and we cite a couple of them. In [16, 17], the authors proved the existence of
radially symmetric solutions concentrating on the spheres, and in [18], there is a
positive bound state solution concentrating on the local minimum of the potential
V . The existence of radial solution was obtained in [5] for the case 3 ≤ p < 5.
In [7], the authors constructed positive radially symmetric bound states of (1.3)
with 1 < p < 11/7. In [6], a related Pohozǎev identity was established and the
authors showed that (1.3) has nontrivial solutions in the case 0 ≤ p < 1 or p ≥ 5.
In [2, 13, 23], the authors proved the existence of infinity many radially symmetric
solutions. Ruiz and Vaira [24] proved the existence of multi-bump solutions whose
bumps concentrating around a local minimum of the potential. Also, there are a lot
of results on Schrödinger-Poisson systems with general classes of nonlinear terms. In
[26], existence and nonexistence nontrivial solutions of Schrödinger-Poisson system
with sign-changing potential were obtained by using variational methods. Sun,
Wu and Feng [27] studied the multiplicity of positive solutions for a nonlinear
Schrödinger-Poisson system when the nonlinearity f (x, u) = Q(x)|u|p−2 u, 2 < p <
6; furthermore, they showed that the number of positive solutions was dependent
of the profile of Q(x). In [28], the authors proved the existence and nonexistence
solutions of Schrödinger-Poisson system with an asymptotically linear nonlinearity.
In [29], existence and multiplicity results were established. We refer to [1, 2, 3, 8,
9, 11, 13, 14, 20, 22] for some more results on this subject.
Recently, semi-classical states for Schrödinger-Poisson systems with much more
general nonlinear term have been object of interest for many authors. Bonheure,
Di Cosmo and Mercuric [11] proved the existence the solutions for the weighted
nonlinear Schrödinger-Poisson systems whose bumps concentrating around a circle. He and Zou [20] showed the existence and concentration of ground states for
Schrödinger-Poisson equations with critical growth.
But, in most of the above papers, the potential V (x) either has a limit at infinity,
or is required to be radial symmetry respect to x. Motivated by some related works,
the aim of this paper is to study the existence of solution of (1.3) concentrating on
a given set of local minima of V (x). We take the penalization arguments going back
to del Pino and Felmer [21] to a wider class of the potentials V (x) and nonlinearity
f (s) ∈ C 1 (R, R).
In this article, we use the following assumptions:
(A1) V (x) ≥ V0 > 0 for all x ∈ R3 .
(A2) f (s) = o(s3 ) as s → 0.
(A3) There exists q ∈ (3, 5) such that lims→+∞ f (s)/sq = 0.
(A4) There exists some 4 < θ < q + 1 such that
Z s
0 < θF (s) = θ
f (t)dt ≤ f (s)s for all s > 0.
0
3
(A5) For all x ∈ R , f (x, s)/s is nondecreasing in s ≥ 0.
The main result of this paper reads as follows.
Theorem 1.1. Assume that (A1)–(A5) hold, and that there is a bounded and compact domain Λ in R3 such that
inf V (x) < min V (x).
x∈Λ
x∈∂Λ
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SEMI-CLASSICAL STATES
3
Then there exists 0 > 0 such that for any ∈ (0, 0 ), problem (1.3) has a positive
solution u . Moreover, u has at most one local (hence global) maximum x ∈ Λ
such that
lim V (x ) = V0 .
→0
Also, there are constants C, c > 0 such that
u (x) ≤ C exp(−c
x − x
).
(1.4)
Remark 1.2. (i) We point out that no restriction on the global behavior of V (x)
is required other than condition (A1). This is an improvement on some previous
works, see, e.g.,[11] [20] and references therein.
(ii) Condition (A5) holds if f (s)/s3 is an increasing function of s > 0. In fact,
that f (s)/s3 is increasing is required in [20].
This article is organized as follows: In section2, influenced by the work of del
Pino and Felmer [21], we introduce a modified functional for any > 0 and show it
has a ground state solution u (x). In Section3, we give the uniform boundedness of
maxx∈∂Λ u (x) and the critical value c when goes to zero. In section4, we show the
critical point of the modified functional which satisfies the original problem (1.3),
and investigate its concentration and exponential decay behavior, which completes
the proof Theorem 1.1.
Hereafter we use the following notation:
• H 1 (R3 ) is usual Sobolev space endowed with the standard scalar product and
norm
Z
Z
(∇u∇v + uv)dx; kuk2 =
(u, v) =
R3
(|∇u|2 + u2 )dx.
R3
• D1,2 (R3 ) is the completion of C0∞ (R3 ) with respect to the norm
Z
kuk2D1,2 (R3 ) =
|∇u|2 dx.
R3
• H −1 denotes the dual space of H 1 (R3 ).
• Lq (Ω), 1 ≤ q ≤ +∞, Ω ⊆ R3 , denotes a Lebesgue space, the norm in Lq (Ω) is
denoted by |u|q,Ω .
• For any R > 0 and for any y ∈ R3 , BR (y) denotes the ball of radius R centered
at y.
• C, c are various positive constants.
• o(1) denotes the quantity which tends to zero as n → ∞.
It is well known that for every u ∈ H 1 (R3 ), the Lax-Milgram theorem implies
that there exists a unique φu ∈ D1,2 (R3 ) such that
Z
Z
∇φu ∇vdx =
u2 dx, ∀v ∈ D1,2 (R3 ),
R3
R3
where φu is a weak solution of −∆φ = u2 with
Z
u2 (y)
dy.
φu (x) =
R3 |x − y|
Substituting φu in (1.3), we can rewrite (1.3) as the equivalent equation
− 2 u + V (x)u + −2 φu u = f (u).
(1.5)
4
H. ZHU
Let
1
3
EJDE-2016/75
Z
H = {u ∈ H (R ) :
V (x)u2 dx < +∞}
R3
be the Sobolev space endowed with the norm
Z
kuk2 =
(2 |∇u|2 + V (x)u2 )dx.
R3
We see that (1.5) is variational and its solutions are the critical points of the functional
Z
Z
Z
1
1
2
2
2
2
( |∇u| + V (x)u )dx + 2
φu (x)u dx −
F (u)dx. (1.6)
I (u) =
2 R3
4 R3
R3
Clearly, under the hypotheses (A2)–(A5), we see that I is well-defined C 1 functional. In the following proposition, we summarize some properties of φu , which
are useful to study our problem.
Proposition 1.3 ([12]). For any u ∈ H 1 (R3 ), we have
(i) φu : H 1 (R3 ) → D1,2 (R3 ) is continuous, and maps bounded sets into bounded
sets;
(ii) if un * u in H 1 (R3 ), thenR φun * φu in D1,2 (R3 );
(iii) φu ≥ 0, kφu kD1,2 (R3 ) , and R3 φu u2 dx ≤ Ckuk4 ;
(iv) φtu (x) = t2 φu for all t ∈ R.
2. Solution of the modified equation
In this section, we find a solution u of problem (1.3) concentrating on a given
set Λ, we modify the nonlinearity f (s). Here we follow an approach used by del
Pino and Felmer [21].
V0
θ
, a > 0 be such that f (a)
Let k > θ−4
a = k , and set
(
f (s), if s ≤ a,
(2.1)
f˜(s) = V0
if s > a,
k s,
and define
g(., s) = χΛ f (s) + (1 − χΛ )f˜(s),
(2.2)
where Λ is the bounded domain as in the assumptions of Theorem1.1, and χΛ
denotes its characteristic function. It is easy to check that g(x, s) satisfies the
following assumptions:
(A6) g(x, s) = o(s3 ) as s → 0.
(A7) There exists q ∈ (3, 5) such that lims→+∞ g(x,s)
= 0.
sq
(A8) There exists a bounded subset K of R3 , int(K) 6= ∅ such that
0 < θG(x, s) ≤ g(x, s)s for all x ∈ K, s > 0,
1
0 ≤ 2G(x, s) ≤ g(x, s)s ≤ V (x)s2 for all s > 0, x ∈ K c .
k
(A9) The function g(x,s)
is increasing for s > 0.
s
Now, we consider the modified equation
−2 ∆u + V (x)u + φ(x)u = g(x, s),
−2 ∆φ = u2 ,
x ∈ R3 ,
(2.3)
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SEMI-CLASSICAL STATES
5
where V satisfies condition (A1), and g satisfies (A6)–(A9). Here we have set = 1
for notational simplicity.
The functional associated with (2.3) is
Z
Z
Z
1
1
2
2
2
J(u) =
(|∇u| + V (x)u )dx +
φu (x)u dx −
G(x, u)dx,
(2.4)
2 R3
4 R3
R3
which is of class C 1 in H with associated norm k · kH .
In the rest of this section, we show some lemmas related to the functional J.
First, we show the functional J satisfying the mountain pass geometry.
Lemma 2.1. The functional J satisfies the following conditions:
(i) There exist α, ρ > 0 such that J(u) ≥ α for all kukH = ρ.
(ii) There exists e ∈ Bρc (0) with J(e) < 0.
Proof. (i) For any u ∈ H\{0} and ε > 0, by (A2) and (A3) there exists C(ε) > 0
such that
|f (s)| ≤ ε|s| + Cε |s|q , ∀s ∈ R.
By the Sobolev embedding H ,→ Lp (R3 ), with p ∈ [2, 6], we have
Z
1
2
[χΛ (x)F (u) + (1 − χΛ (x))F̃ (u)]dx
J(u) ≥ kukH −
2
3
ZR
1
≥ kuk2H −
F (u)dx
2
R3
Z
1
ε
Cε
≥ kuk2H −
|u|2 dx −
|u|p+1 dx
2
2 R3
q+1
1
≥ kuk2H − C1 εkuk2H − C2 Cε kukp+1
H .
2
Since ε is arbitrarily small, we can choose constants α, ρ such that J(u) ≥ ρ > 0
for all kukH = ρ.
(ii) By (A4), we have F (s) ≥ Csθ − C for all t > 0, Choosing u ∈ H\{0} not
negative, with its support contained in the set K, we see that
Z
Z
t2
t4
2
2
J(tu) = kukH +
φu u dx −
G(x, tu)dx
2
4 R3
R3
Z
Z
t2
t4
2
2
θ
≤ kukH +
φu u dx − Ct
uθ dx + C|K| < 0
2
4 R3
K
for some t > large enough. So, we can choose e = t∗ u for some t∗ > 0, and (ii)
follows.
By lemma 2.1 and the mountain pass theorem, there is a (P S)c sequence {un } ⊂
H such that J(un ) → c in H −1 with the minmax value
c = inf max J(γ(t)),
γ∈Γ 0≤t≤1
(2.5)
where
Γ := {γ ∈ C([0, 1], H) : γ(0) = 0, J(γ(1)) < 0}.
Lemma 2.2. Let {un } ⊂ H be a (P S)c sequence for c > 0. Then un has a
convergent subsequence.
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H. ZHU
EJDE-2016/75
Proof. First, we show that {un } is bounded in H. In fact, using (A8) we easily see
that
Z
Z
Z
2
2
2
(|∇un | + V (x)un )dx +
φun un dx ≥
g(x, un )un dx + o(kun kH ). (2.6)
R3
R3
K
Z
Z
1
1
(|∇un |2 + V (x)u2n )dx +
φu u2 dx
2 R3
4 R3 n n
Z
=
G(x, un )dx + O(1)
(2.7)
3
ZR
Z
1
≤
G(x, un )dx +
V (x)u2n dx + O(1)
2k K c
K
Thus, from (2.6) (2.7) and (A8) we find
Z
Z
2
2
2
V (x)un dx + o(kun kH ) + O(1) ≥ (1 − )
(|∇un |2 + V (x)u2n )dx.
k Kc
k R3
(2.8)
Then, it follows from the choice of k in (A8) that {un } is bounded in H.
Then there is a subsequence, still denoted by {un } such that un * u weakly in
H. We now prove this convergence is actually strong. In deed, it suffices to show
that, given δ > 0, there is an R > 0 such that
Z
lim sup
(|∇un |2 + V (x)u2n )dx ≤ δ.
(2.9)
n→∞
c
BR
Let ξR (x) ∈ C ∞ (R3 , R) be a cut-off function such that 0 ≤ ξR ≤ 1 and
(
0 for |x| ≤ R2 ,
ξR (x) =
1, for |x| ≥ R
3
and |∇ξR (x)| ≤ C
R for all x ∈ R . Moreover we may assume that R is chosen so
that K ⊂ B R . Since {un } is a bounded (P S)c sequence, we have that
2
hJ 0 (un ), ξR un i = o(1),
(2.10)
so that
Z
Z
Z
(|∇un |2 + V (x)u2n )dx +
un ∇un ∇ξR dx +
φun un ξR dx
R3
R3
R3
Z
Z
1
=
g(x, un )un ξR dx + o(1) ≤
V (x)u2n ξR dx + o(1).
k
3
R
R3
(2.11)
We conclude that
Z
c
BR
V (x)u2n dx ≤
C
kun kL2 (R3 ) k∇un kL2 (R3 ) ,
R
from where (2.9) follows.
(2.12)
Lemma 2.1 implies that c defined in (2.5) is a critical value of J.
Remark 2.3. Similar to the proof of lemma 2.2, it is not difficult to see that c can
be characterized as
c = inf sup J(tu).
u∈H\{0} t≥0
Since the modified function g satisfies assumptions (A6)–(A9), the results of the
above yield the following lemma.
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SEMI-CLASSICAL STATES
7
Lemma 2.4. For any > 0, there exists a critical point for J at level
J (u ) = c = inf max J (γ(t)),
γ ∈Γ 0≤t≤1
(2.13)
where
1
J (u) =
2
Γ := {γ ∈ C([0, 1], H) : γ(0) = 0, J (γ(1)) < 0},
Z
Z
Z
1
(2 |∇u|2 + V (x)u2 )dx + 2
φu u2 dx −
G(x, u)dx.
4 R3
R3
R3
(2.14)
3. Some estimates
To show that the solution u found in lemma 2.4 satisfies the original problem
and concentrates at some point in Λ, we need to study the behavior of u as → 0.
We begin with an energy estimate.
Proposition 3.1 (Upper estimate of the critical value). For small enough, the
critical value c defined (2.13) satisfies
c = J (u ) ≤ 3 (c0 + o(1))
as → 0.
Moreover, there exists C > 0 such that
Z
(2 |∇u |2 + V (x)u2 )dx ≤ C3 .
(3.1)
(3.2)
R3
Proof. Set V0 = minΛ V = V (x0 ), and let
c0 := inf max I0 (γ(t)),
γ∈Γ 0≤t≤1
(3.3)
where
Z
Z
Z
1
1
2
2
2
(|∇u| + V0 u )dx +
φu u dx −
F (u)dx,
I0 (u) =
2 R3
4 R3
R3
Γ := {γ ∈ C([0, 1], H 1 (R3 )) : γ(0) = 0, I0 (γ(1)) < 0}.
From (3.3), for any δ > 0, there exists a continuous path γδ : [0, 1] → H 1 (R3 ) such
that γδ (0) = 0, I0 (γδ (1)) < 0 and
c0 ≤ max I0 (γδ (t)) ≤ c0 + δ.
0≤t≤1
Let η be a smooth cut-off function with support in Λ such that 0 ≤ η ≤ 1, η = 1 in
a neighborhood of x0 and |∇η| ≤ C. We consider the path
x − x0
γ¯δ (t)(x) = η(x)γδ (t)(
).
Setting
x − x0
γ¯δ (t)(x) := vt (
),
we compute, by a charge of variable
Z
Z
1
[2 |∇γ¯δ (t)|2 + V (x)γ¯δ (t)2 ]dx −
G(x, γ¯δ (t))dx
2 R3
R3
Z
(3.4)
31
2
2
3
= [|∇vt (x)| + V (x0 + x)vt (x)]dx − G(x0 + x, vt (x))dx.
2
R3
The Hardy-Littlewood Sobolev inequality leads to
Z
Z hZ
i
γ¯δ (t)(y)2
2
γ¯δ (t)dy dx
φγ¯δ (t)(x) γ¯δ (t)(x) dx =
R3
R3
R3 |x − y|
8
H. ZHU
= 5
= 5
EJDE-2016/75
Z
ZR
Z
3
R3
R3
vt2 (x)vt2 (y)
dx dy
|x − y|
φvt vt2 dx.
For small enough, we obtain
−3 J (γ¯δ (t)) → I0 (γδ (t)) + o(1).
It follows that small enough, γ¯δ belongs to the class of paths Γ defined by (2.14).
We deduce that
−3 c ≤ −3 max J (γ¯δ (t)) → max I0 (γ¯δ (t)) + o(1) ≤ (c0 + δ) + o(1).
0≤t≤1
0≤t≤1
Since δ > 0 is arbitrary, (3.1) is proved.
Z
Z
Z
1
1
2
2
2
2
( |∇u | + V (x)u )dx + 2
φu u dx −
G(x, u )dx
J (u ) =
2 R3
4 R3 3
Z
Z
ZR
1
1
=
(2 |∇u |2 + V (x)u2 )dx + 2
φu u2 dx −
G(x, u )dx
2 R3
4 R3 K
Z
−
G(x, u )dx
(3.5)
R3 \{K}
Z
Z
1
1
≥
(2 |∇u |2 + V (x)u2 )dx −
V (x)u2 dx
4 R3
2k R3
Z
1
1
)
(2 |∇u |2 + V (x)u2 )dx,
≥( −
4 2k R3
1
where C = 41 − 2k
> 0 thanks to the choice of k. Combining (3.1) and (3.5), it is
easy to obtain (3.2).
Next, we give a proposition that is the crucial step in the proof of Theorem 1.1.
Proposition 3.2.
lim max u (x) = 0.
→0 ∂Λ
(3.6)
Moreover, for all sufficiently small enough, u possesses one local maximum x ∈ Λ
and we must have
lim V (x ) = V0 = min V (x).
(3.7)
→0
x∈Λ
Proof. To prove this proposition we establish that If n → 0 and xn ∈ Λ are such
that un ≥ b > 0, then
lim V (xn ) = V0 .
(3.8)
n→∞
We take three steps to prove this claim.
Step1: We argue by contradiction. Thus we assume, passing to a subsequence,
that xn → x∗ ∈ Λ̄ and
V (x∗ ) > V0 .
(3.9)
We consider the sequence vn (x) = un (xn + n x). A simple computation shows
2 φvn (x) = φun (xn + n x).
The function vn satisfies the equation
− ∆vn + V (xn + n x)vn + φvn vn = g(xn + n x, vn ), x ∈ Ωn ,
(3.10)
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SEMI-CLASSICAL STATES
9
where Ωn = −1
n {H − xn }. As a consequence of (3.2), we see that vn is bounded in
H 1 (R3 ), and from elliptic estimates, we deduce that there exists v ∈ H 1 (R3 ) such
that
2
vn → v in Cloc
(R3 ).
Let χn (x) = χΛ (xn + n x), then χn (x) * χ in any Lp (R3 ) over compacts with
0 ≤ χ ≤ 1. Now, we claim that
Z
Z
φvn vn ϕdx →
φv vϕdx for any ϕ ∈ C0∞ (R3 ).
R3
R3
In fact, we can assume support ϕ ⊂ Ω, where Ω is a bounded domain. Then
Z
Z
|
φvn vn ϕdx −
φv vϕdx|
R3
R3
Z
Z
=|
φvn (vn − v)ϕdx +
(φvn − φv )vϕdx|
3
R3
ZR
Z
≤|
φvn (vn − v)ϕdx| + |
(φvn − φv )vϕdx|
R3
R3
≤ |φvn |L6 (Ω) |vn − v|L2 (Ω) |ϕ|L3 (Ω) + o(1) → o(1).
Therefore, v satisfies the limiting equation
− ∆v + V (x∗ )v + φv v = ḡ(x, v),
(3.11)
where
ḡ(x, s) = χ(x)f (s) + (1 − χ(x))f˜(s).
Associated with (3.11) we have functional J¯ : H 1 (R3 ) → R defined as
Z
Z
1
1
¯
[|∇u|2 + V (x∗ )u2 ]dx +
φu u2 dx
J(u)
=
2 R3
4 R3
Z
−
Ḡ(x, u)dx, u ∈ H 1 (R3 ),
(3.12)
R3
Rs
¯ Set
ḡ(x, t)dt. Then v is a critical point of J.
Z
Z
1
1
Jn (u) =
[|∇u|2 + V (xn + n x)u2 ]dx +
φu u2 dx
2 Ωn
4 Ωn
Z
−
G(xn + n x, u)dx, u ∈ H01 (Ωn ).
where Ḡ(x, s) =
0
Ωn
−3
n Jn (un ).
Then Jn (vn ) =
So the key step in the proof of proposition is the
following step.
¯
¯
Step2: lim inf n→∞ Jn (vn ) ≥ J(v).
In particular, J(v)
≤ c0 , where c0 is given by
(3.3).
Proof: Write
1
1
hn = [|∇vn |2 + V (xn + n x)vn2 ] + φvn vn2 − Ḡ(xn + n x, u).
2
4
Then, choose R > 0, since vn converges in the C 1 sense over compacts to v, we
have
Z
Z
Z
Z
1
1
lim
hn dx =
[|∇v|2 + V (x∗ )v 2 ]dx +
φv v 2 dx −
Ḡ(x, v)dx.
n→∞ B
2 BR
4 BR
BR
R
10
H. ZHU
EJDE-2016/75
Since v ∈ H 1 (R3 ), for each δ > 0, we have
Z
¯ − δ,
lim
hn dx ≥ J(v)
n→∞
(3.13)
BR
provided that R was chosen sufficiently large. Then it only suffices to check that
for large enough R
Z
lim
hn dx ≥ −δ.
(3.14)
n→∞
For any fixed R > 0, let ξR (x) ∈
Ωn \BR
∞
C0 (R3 , R)
(
ξR (x) =
be a cut-off function such that
0 for |x| ≤ R − 1,
1, for |x| ≥ R,
3
and |∇ξR (x)| ≤ C
R for all x ∈ R and C > 0 is a constant.
1
We use wn = ξR vn ∈ H (Ωn ) as a test function for Jn0 (vn ) = 0 to obtain
Z
Z
0 = Jn0 (vn )wn = En +
(2hn + gn )dx +
φvn vn2 dx
Ωn \BR
Ωn \BR
Z
Z
(3.15)
≤ En +
2hn dx +
φvn vn2 dx,
Ωn \BR
Ωn \BR
where gn = 2G(xn + n x, vn ) − g(xn + n x, vn )vn , and En is given by
Z
En =
[∇vn ∇(ξR vn ) + V (xn + n x)ξR vn2 + φvn vn2 ξR ]dx
B \B
Z R R−1
=
g(xn + n x, vn )ξR vn dx.
(3.16)
BR \BR−1
R
Since vn is bounded in H 1 (R3 ), it follows that R3 φvn vn2 dx ≤ Ckun k4 . The fact
that v ∈ H 1 (R3 ) implies that for given δ > 0, there exists R > 0 sufficiently large
such that
Z
lim |En | ≤ δ,
φvn vn2 dx ≤ δ.
n→∞
Ωn \BR
On the other hand, the definition of g together with the properties of f give that
gn ≤ 0. Using this in (3.15), (3.14) follows, and the proof of step2 is complete.
Step3: Now, we are ready to obtain a contradiction with (3.8). Since v is a critical
¯ and ḡ satisfies (A9), we have that
point of J,
¯ = max J(tv).
¯
J(v)
(3.17)
t>0
Then since f (s) ≥ f˜(s) for all s > 0 we have
¯ ≥
J(v)
inf3
sup I ∗ (τ u)∆qc∗ ,
1
(3.18)
u∈H (R )\{0} τ >0
where
I ∗ (u) =
1
2
Z
R3
[|∇u|2 + V (x∗ )u2 ]dx +
1
4
Z
R3
φu u2 dx −
Z
F (u)dx.
(3.19)
R3
¯ > c∗ , which contradicts step 2,
But, since V (x∗ ) > V0 , we have c∗ > c0 ; hence J(v)
and the proof of the claim, i.e. (3.8) is follows.
To conclude the proof of proposition 3.2, we show that u has at most one
maximum point in Λ. The proofs rely on the the arguments carried out in step2
and so we sketch it. By contradiction, assume that, the existence of sequence n → 0
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SEMI-CLASSICAL STATES
11
such that un has two distinct maxima x1n and x2n in Λ. Set vn (x) = un (x1n + n x),
2
1
and it is easy to check that −1
n (xn − xn ) is a maximum point of vn (x), two cases
occur.
2
1
Case 1: −1
n (xn − xn ) is bounded. From (3.2) and elliptic estimates, up to a
subsequence, vn → v uniformly over compacts, where v ∈ H 1 (R3 ) maximizes at zero
2
1
and solves −∆v + V (x1 )v + φv v = f (v), here x1 = limn→∞ x1n . Since −1
n (xn − xn )
3
is bounded and hence, up to a subsequence, it converges to some p ∈ R . So we
2
1
conclude that p = 0; therefore −1
n (xn − xn ) ∈ Br for n large enough, which is
impossible since 0 is the only critical point of v in Br .
2
1
2
Case2: −1
n (xn − xn ) is unbounded. Let v˜n (x) = un (n x + xn ), then there exists
2
ṽ such that ṽ is the solution of −∆v + V (x )v + φv v = f (v), here x2 = limn→∞ x2n .
2
1
Note that |−1
n (xn − xn )| → +∞, then for any R > 0 the balls B̃R ∩ B̄ = ∅, where
2
1
B̄ = B̃R (−1
n (xn − xn )), repeat the arguments of step2, we find that for any ν > 0
it is possible to choose that R > 0 large enough such that
Z
˜ − ν,
lim
hn dx ≥ J(ṽ)
(3.20)
n→∞
B̄ where
1
˜
J(u)
=
2
Z
(|∇u|2 + V (x2 )u2 )dx +
R3
and
1
4
Z
φu u2 dx −
R3
Z
F (u)dx,
R3
Z
hn dx ≥ −ν.
lim
n→∞
(3.21)
R3 \(BR ∪B )
Similar to the argument in (3.13), we obtain
Z
hn dx ≥ J1 (v) − δ,
lim
n→∞
(3.22)
BR
where
1
J1 (u) =
2
Z
1
(|∇u| + V (x )u )dx +
4
R3
2
1
2
Z
2
Z
φu u dx −
R3
F (u)dx.
R3
From (3.22),(3.20) and (3.21) we conclude that
Z
˜ − 3ν.
lim
hn dx ≥ J1 (v) + J(ṽ)
n→∞
(3.23)
R3
Since ν is arbitrary we find that
˜
−3
n Jn (un ) = lim Jn (vn ) ≥ J1 (v) + J(ṽ) ≥ 2c0 ,
n→∞
which contradicts (3.1). The proof of proposition 3.2 is now complete.
4. Proof of Theorem 1.1
In this section, we shall prove the existence, concentration, and exponential decay
of ground state solution of (1.3) for small .
Proof of Theorem 1.1. By proposition 3.2, there exists 0 such that for 0 < < 0 ,
u (x) < a for all x ∈ ∂Λ.
(4.1)
The function u ∈ H solves the equation
− 2 ∆u + V (x)u + −2 φu u = g(x, u).
(4.2)
12
H. ZHU
EJDE-2016/75
Choose (u − a)+ as a test function in (4.2), after integration by parts one gets
Z
h
2 |∇(u − a)+ |2 + c(x)(u − a)2+ + −2 φu u (u − a)+
3
R \{Λ}
(4.3)
i
+ c(x)a(u − a)+ dx = 0,
where
g(x, u (x))
.
u (x)
The definition of g yields that c(x) > 0 in R3 \{Λ}, hence all terms in (4.3) are zero.
We conclude in particular
c(x) = V (x) −
u (x) ≤ a for all R3 \{Λ}.
Consequently, u is a solution to equation (1.3), and by proposition 3.2, we know
that the maximum value of u is achieved at a point x ∈ Λ and it is away from
zero. To obtain (1.4), we need the following proposition, which is a very particular
version of [15, Theorem 8.17].
Proposition 4.1 ([15]). Suppose that t > 3, h ∈ Lt/2 (Ω) and u ∈ H 1 (Ω) satisfies
in the weak sense
−∆u ≤ h(x) in Ω,
3
where Ω is an open subset of R . Then, for any ball B2R (y) ⊂ Ω,
sup
x∈BR (y)
u(x) ≤ C(ku+ kL2 (B2R (y)) + khkLt/2 (B2R (y)) ),
where C depends on t and R.
Lemma 4.2. Let v (x) = u (x + x), where x is the unique maximum of u , then
there exists ∗ > 0 such that lim|x|→∞ v (x) = 0 uniformly on ∈ (0, ∗ ).
Proof. Since u (x) is the solution of (1.3), by (3.2) then
kv kH ≤ C,
(4.4)
and also v (x) satisfies
−∆v + V (x + x)v (x) + φv v = f (v ).
Now, for any sequence n → 0, there is a subsequence such that
xn → x̄; V (x̄) = V0 .
From (4.4) and elliptic estimates, we know that this subsequence can be chosen in
such a way that vn → v uniformly over compacts, where v ∈ H 1 (R3 ) solves
− ∆v + V0 v + φv = f (v).
(4.5)
1
3
→ v ∈ H (R ). Since f˜(s) ≤ f (s) for all s ≥ 0, by (3.1)
Next, we prove that vn
we have
In (vn ) ≤ −3
n Jn (un ) ≤ c0 ,
where
In (u) =
1
2
Z
[|∇u|2 + V (xn + n x)u2 ]dx +
Ωn
Z
−
F (xn + n x, u)dx, Ωn
Ωn
3
= −1
n {R − xn }.
1
4
Z
φu u2 dx
Ωn
(4.6)
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SEMI-CLASSICAL STATES
13
On the other hand, using Fatou’s lemma and the weak limit of vn ,
1
In (vn ) = In (vn ) − hIn0 (vn ), vn i
4
Z
Z
1
1
2
2
=
[|∇vn | + V (xn + n x)vn ]dx +
[f (vn )vn − 4F (vn )]dx
4 Ωn
4 Ωn
Z
Z
1
1
[|∇vn |2 + V0 v2n ]dx +
[f (vn )vn − 4F (vn )]dx
≥
4 Ωn
4 Ωn
Z
Z
1
1
≥
[|∇v|2 + V0 v 2 ]dx +
[f (v)v − 4F (v)]dx
4 R3
4 R3
1
= I0 (v) − hI00 (v), vi ≥ c0 .
4
So, In (vn ) → c0 as n → ∞, and it is easy to verify from the above inequalities,
Z
Z
2
2
lim
(|∇vn | + V0 vn )dx =
(|∇v|2 + V0 v 2 )dx.
n→∞
R3
R3
1
Therefore, using that vn * v weakly in H (R ), we conclude vn → v in H 1 (R3 ).
As a consequence of the above limit, we have
Z
lim
v2n dx = 0.
(4.7)
R→∞
3
|x|≥R
Applying proposition 4.1 in the inequality
−∆vn ≤ −∆vn + V (n x + xn )vn + φvn vn = hn (x)∆qf (vn )
in R3 ,
we have that for some t > 3, khn k 2t ≤ C for all n. Moreover,
sup
x∈BR (y)
vn (x) ≤ C(kvn kL2 (B2R (y)) + khn kLt/2 (B2R (y)) )
for all y ∈ R3 ,
which implies that kvn kL∞ (R3 ) is uniformly bounded. Then by (4.7), we have
lim vn (x) = 0
|x|→∞
uniformly on n ∈ N.
Consequently, there exists ∗ > 0 such that
lim v (x) = 0
|x|→∞
uniformly on ∈ (0, ∗ ) .
To show the exponential decay of u , we only need the following result involving
of v .
Lemma 4.3. There exist constants C > 0 and c > 0 such that
v (x) ≤ Ce−c|x|
for all x ∈ R3 .
Proof. By lemma 4.2 and (A2), there exists R1 > 0 such that
V0
f (v (x))
≤
v (x)
2
for all |x| ≥ R1 , ∈ (0, ∗ ).
(4.8)
Fix ω(x) = Ce−c|x| with c2 < V0 /2 and Ce−cR1 ≥ v for all |x| = R1 . It is easy to
check that
∆ω ≤ c2 ω for all |x| =
6 0.
(4.9)
14
H. ZHU
EJDE-2016/75
So
V0
v
2
Define ω = ω − v . Using (4.10) and (4.9), we obtain
− ∆v + V0 v ≤ −∆v + V0 v + φv v = f (v ) ≤
−∆ω +
V0
ω ≥ 0,
2
in |x| ≤ R1 , ω ≥ 0,
for all |x| > R1 .
(4.10)
on |x| = R1 , lim ω (x) = 0.
|x|→∞
The classical maximum principle implies that ω ≥ 0 in |x| ≥ R1 and by the work
in [19], we conclude that
v (x) ≤ Ce−c|x|
for all |x| ≥ R1 , ∈ (0, ∗ ).
By the definition of v and lemma 4.3, we have
|x − x |
x − x
) ≤ Cexp(−c
)
for all x ∈ R3 , ∈ (0, ∗ ). The proof of Theorem 1.1 is complete.
u (x) = v (
Acknowledgments. This work was supported by Natural Science Foundation of
China (11201083) and China Postdoctoral Science Foundation (2013M542038).
The author would like to express her sincere thanks to the anonymous referees
for their careful reading of the manuscript and valuable comments and suggestions.
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Hongbo Zhu
School of Mathematics and Statistics, Central China Normal University, Wuhan 430079,
China.
School of Applied Mathematics, Guangdong University of Technology, Guangzhou
510006, China
E-mail address: zhbxw@126.com