Table Of ContentExistence of positive multi-bump solutions for a
Schrödinger-Poisson system in R3
Claudianor O. Alvesa∗, Minbo Yangb†
5
1 a. Universidade Federal de Campina Grande
0 Unidade Acadêmica de Matemática
2
CEP: 58429-900,Campina Grande - Pb, Brazil
n
b. Department of Mathematics, Zhejiang Normal University
a
J Jinhua, 321004,P. R. China.
3
1
]
P
Abstract
A
In this paper we are going to study a class of Schrödinger-Poissonsystem
.
h
t −∆u+(λa(x)+1)u+φu=f(u) in R3,
a
m −∆φ=u2 in R3.
(cid:26)
[
Assuming that the nonnegative function a(x) has a potential well int(a−1({0}))
1 consisting of k disjoint components Ω1,Ω2,.....,Ωk and the nonlinearity f(t) has a
v
subcritical growth, we are able to establish the existence of positive multi-bump
0
solutions by variational methods.
3
9
Mathematics Subject Classifications (2010): 35J20, 35J65
2
0
. Keywords: Schrödinger-Poissonsystem, multi-bump solution, variational methods.
1
0
5 1 Introduction
1
:
v
This paper was motivated by some recent works concerning the nonlinear Schrödinger-
i
X
Poisson system
r
a
−i∂ψ = −∆ψ+V(x)ψ+φ(x)ψ−|ψ|p−2ψ in R3,
∂t (NSP)
−∆φ= |ψ|2 in R3,
(cid:26)
whereV :R3 → Risanonnegativecontinuousfunctionwith inf V(x) >0,2 < p < 2∗ = 6
x∈R3
and ψ : Ω → C and φ: Ω→ R are two unknown functions.
The first equation in system (NSP), called Schrödinger equation, describes quantum
(non-relativistic) particles interacting with the eletromagnetic field generated by the
motion. Aninteresting phenomenon ofthis Schrödinger type equation isthatthe potential
∗C. O. Alves was partially supported by CNPq/Brazil 301807/2013-2 and INCT-MAT,
[email protected]
†M. Yang was supported by NSFC(11101374, 11271331), [email protected]
1
φ(x) is determined by the charge of wave function itself, that means, φ(x) satisfies the
second equation (Poisson equation) in system (NSP).
As we all know, the knowledge of the solutions for the elliptic system
−∆u+V(x)u+φu = f(u) in R3,
(SP)
−∆φ= u2 in R3,
(cid:26)
has a great importance in studying stationary solutions ψ(x,t) = e−itu(x) of (NSP). It
is convenient to observe that the system (SP) contains two kinds of nonlinearities. The
first one is φ(x)u which is nonlocal, since the electrostatic potential φ(x) depends also
on the wave function, and is used to describe the interaction between the solitary wave
and the electric field. The second type of nonlinearity f(u) is a local one which has been
used to model the external forces involving only functions of fields. For more information
about the physical background of system (SP), we cite the papers of Benci-Fortunato [6],
Bokanowski & Mauser [8], Mauser [24], Ruiz [26], Ambrosetti-Ruiz [4] and S’anchez &
Soler [28].
Animportant factforsystem(SP)isthatitcanbereducedintoonesingleSchrödinger
equation with a nonlocal term (see, for instance, [5, 17, 26, 30]). Effectively, by the Lax-
Milgram Theorem, given u ∈ H1(R3), there exists an unique φ = φ ∈ D1,2(R3) such
u
that
−∆φ= u2 in R3.
Byusingstandardarguments, weknowthatφ verifiesthefollowing properties(foraproof
u
see [11, 26, 30]):
Lemma 1.1. For any u ∈ H1(R3), we have
u2(y)
i) φ (x) = dy for all x ∈ R3;
u
R3 |x−y|
Z
ii) there exists C > 0
|∇φ |2dx = φ u2dx ≤ C||u||4 ∀u∈ H1(R3),
u u H1(R3)
R3 R3
Z Z
1
2
where ||u||H1(R3) = (|∇u|2+|u|2)dx .
R3
(cid:18)Z (cid:19)
iii) φ ≥ 0 ∀u∈ H1(R3);
u
iv) φ = t2φ , ∀t> 0 and u∈ H1(R3);
tu u
v) if u ⇀ u in H1(R3), then φ ⇀ φ in D1,2(R3) and
n un u
lim φ u2dx ≥ φ u2dx.
n→+∞ R3 un n R3 u
Z Z
vi) if u → u in H1(R3), then φ → φ in D1,2(R3). Hence,
n un u
lim φ u2dx = φ u2dx.
n→+∞ R3 un n R3 u
Z Z
2
Therefore, (u,φ) ∈ H1(R3)×D1,2(R3) is a solution of (SP) if, and only if, u∈ H1(R3)
is a solution of the nonlocal problem
−∆u+V(x)u+φ u= f(u) in R3,
u (P)
u ∈H1(R3),
(cid:26)
where φ = φ∈ D1,2(R3).
u
Now, we would like to emphasize that the existence of solutions for problem (P) can
be established via variational methods. Associated to the elliptic equation (P), we have
the energy functional I : E → R given by
1 1
I(u) = ||u||2+ φ u2dx− F(u)dx,
u
2 4 R3 R3
Z Z
s
where F(s) = f(t)dt and E is the function space
0
R
E = u∈ H1(R3) : V(x)|u|2 < +∞
R3
(cid:26) Z (cid:27)
endowed with the norm
1
2
kuk = (|∇u|2+V(x)|u|2)dx .
R3
(cid:18)Z (cid:19)
Supposingsomeconditionsonf,Lemma1.1impliesthatthefunctionalI iswelldefined
and I ∈ C1(E,R) with
I′(u)v = ∇u∇vdx+ V(x)uvdx+ φ uvdx− f(u)vdx ∀u,v ∈ E.
u
R3 R3 R3 R3
Z Z Z Z
Hence, thecriticalpointsoffunctional I areinfacttheweak solutionsfornonlocalproblem
(P).
From the above commentaries, we know that the system (SP) has a nontrivial solution
if, and only if, the nonlocal problem (P) has a nontrivial solution. In the last years, many
authors had studied the system (SP) and focused their attentions to establish existence
and nonexistence of solutions, multiplicity of solutions, ground state solutions, radial and
nonradial solutions, semiclassical limit and concentrations of solutions, seefor example the
papers of Azzollini & Pomponio [5], Cerami & Vaira [9], Coclite [10], D’Aprile & Mugnai
[11, 12], d’Avenia [13], Ianni [19], Kikuchi [18], and Zhao & Zhao [30]. For the problem set
in a bounded domain, we would like to cite the papers of Siciliano [17], Ruiz & Siciliano
[27] and Pisani & Siciliano [25] for nonnegative solutions and Alves & Souto [3], Ianni
[20] and Kim & Seok [22] for sign-changing solutions. However, related to the existence
of multi-bump solutions for Schrödinger-Poisson system with potential well, as far as we
know, there seems to be no existing results.
In the present paper, we will assume that potential V(x) is of the form
V(x)= λa(x)+1,
3
where λ is a positive parameter and a : R3 → R is a nonnegative continuous function.
Hence, the problems (SP) and (P) can be written respectively as
−∆u+(λa(x)+1)u+φu= f(u) in R3,
(SP)
−∆φ= u2 in R3, λ
(cid:26)
and
−∆u+(λa(x)+1)u+φ u= f(u) in R3,
u (P)
u∈ H1(R3). λ
(cid:26)
To state the main result, we assume that the function a(x) verifying the following
conditions:
(a )Thesetint(a−1({0}))isnonemptyandtherearedisjointopencomponentsΩ ,Ω ,.....,Ω
1 1 2 k
such that
int(a−1({0})) = ∪k Ω (1.1)
j=1 j
and
dist(Ω ,Ω ) > 0 for i 6= j, i,j = 1,2,··· ,k. (1.2)
i j
From (a ), we see that
1
a−1({0}) = ∪k Ω . (1.3)
j=1 j
Related to the function f, we will assume the ensuing conditions:
f(s)
(f ) lim = 0,
1
s→0 s
f(s)
(f ) lim = 0,
2 |s|→+∞ s5
(f ) There exists θ >4 such that
3
0< θF(s)≤ sf(s) ∀s∈ R\{0}.
Moreover, we also assume that the nonlinearity f satisfies
f(s)
(f ) is increasing in |s|> 0.
4 s3
The motivation to investigate problem (SP) goes back to the papers [1] and [15]. In
λ
[15], inspired by [14] and [29], the authors considered the existence of positive multi-bump
solution for the problem
−∆u+(λa(x)+Z(x))u = uq in RN,
(1.4)
u∈ H1(RN),
(cid:26)
q ∈ (1, N+2) if N ≥ 3; q ∈(1,∞) if N = 1,2. The authors showed that the above problem
N−2
has at least 2k −1 solutions u for large values of λ. More precisely, for each non-empty
λ
subset Υ of {1,...,k}, it was proved that, for any sequence λ → ∞ we can extract a
n
4
subsequence (λ ) such that (u ) converges strongly in H1 RN to a function u, which
ni λni
satisfies u= 0 outside Ω = Ω and u , j ∈ Υ, is a least energy solution for
Υ j∈Υ j |Ωj (cid:0) (cid:1)
S
−∆u+Z(x)u = uq, in Ω ,
j
(1.5)
(u∈ H01 Ωj , u > 0, in Ωj.
(cid:0) (cid:1)
After, in [1], Alves extended the results described above to the quasilinear Schrödinger
equation driven by p-Laplacian operator.
Involving the Schrödinger-Poisson system with potential wells, there are not so many
existingpapers. Asfarasweknow,theonlypaperthatconsideredtheexistenceofsolutions
forsystem(SP) isduetoJiangandZhou[21]wheretheauthorsstudiedtheexistenceand
λ
properties of the solutions depending on some parameters. However, nothing is known for
the existence of multi-bump type solutions. Motivated by the above references, we intend
in the present paper to study the existence of positive multi-bump solution for (SP) .
λ
However, we need to point out some difficulties involving this subject:
1-Itiswellknownthattheequation(1.5)playstheroleoflimitequationfor(1.4)asλgoes
to infinity and the ground state solution of (1.5) plays an important role in building the
multi-bump solutions for (1.4). However, little is known about what is the corresponding
limit equation for equation (P) when the parameter λ goes to infinity.
λ
2- Once discovered the limit problem for equation (P) , it is crucial to prove that it has
λ
a specially shaped least energy solution on a subset of the Nehari manifold , see Section 2
for more details.
3- When we apply variational methods to prove the existence of solution to (SP) , we are
λ
led to study a nonlocal, see problem (P) above. However, for this class of problem, it is
λ
necessary to make a careful revision in the sets used in the deformation lemma found in
[1] and [15] to get multi-bump solution, since they don’t work well for this class of system,
see Sections 6 and 7 for more details.
Our main result is the following
Theorem 1.2. Assume that (a ) and (f ) − (f ) hold. Then, there exist λ > 0 with
1 1 4 0
the following property: for any non-empty subset Υ of {1,2,...,k} and λ ≥ λ , problem
0
P has a positive solution u . Moreover, if we fix the subset Υ, then for any sequence
λ λ
λ → ∞ we can extract a subsequence (λ ) such that (u ) converges strongly in H1(R3)
(cid:0)n (cid:1) ni λni
to a function u, which satisfies u = 0 outside Ω = ∪ Ω , and u is a least energy
Υ j∈Υ j |ΩΥ
solution for the nonlocal problem
u2(y)
−∆u+u+ dy u= f(u) in Ω ,
Υ
|x−y|
(cid:18)ZΩΥ (cid:19) (P)∞,Υ
u(x) > 0 ∀x∈ Ω and ∀j ∈ Υ,
j
u∈ H1(Ω ).
0 Υ
5
In the proof of Theorem 1.2, we need to study the existence of least energy solution
for problem (P) . The main idea is to prove that the energy function J associated with
∞,Υ
nonlocal problem (P) given by
∞,Υ
1 1
J(u) = (|∇u|2 +|u|2)dx+ φ u2dx− F(u)dx,
u
2 4
ZΩΥ ZΩΥ ZΩΥ
assumes a minimum value on the set
M = {u∈ N : J′(u)u = 0 and u 6= 0 ∀j ∈ Υ}
Υ Υ j j
where u = u and N is the corresponding Nehari manifold defined by
j |Ωj Υ
N = {u ∈ H1(Ω )\{0} : J′(u)u = 0}.
Υ 0 Υ
More precisely, we will prove that there is w ∈ M such that
Υ
J(w) = inf J(u).
u∈MΥ
After, we use a deformation lemma to prove that w is a critical point of J, and so, w is a
least energy solution for (P) . The main feature of the least energy solution w is that
∞,Υ
w(x) > 0 ∀x ∈ Ω and ∀j ∈ Υ which will be used to describe the existence of multi-bump
j
solutions.
Since we intend to look for positive solutions, through this paper we assume that
f(s)= 0, s ≤ 0.
2 The problem (P)
∞,Υ
Inwhat follows, to show in details the idea of proving the existence of leastenergy solution
for (P) , we will consider Υ = {1,2}. Moreover, we will denote by Ω, N and M the
∞,Υ
sets Ω , N and M respectively. Thereby,
Υ Υ Υ
Ω = Ω ∪Ω ,
1 2
N = {u ∈H1(Ω)\{0} : J′(u)u = 0}
0
and
M = {u∈ N :J′(u)u = J′(u)u = 0 and u ,u 6= 0},
1 2 1 2
with u = u , j = 1,2.
j |Ωj
Since we want to look for least energy for (P) , our goal is to prove the existence of
∞,Υ
a critical point for J in the set M.
6
2.1 Technical lemmas
In what follows, we will denote by || ||, || || and || || the norms in H1(Ω), H1(Ω ) and
1 2 0 0 1
H1(Ω ) given by
0 2
1
2
||u|| = (|∇u|2+|u|2)dx ,
(cid:18)ZΩ (cid:19)
1
2
||u|| = (|∇u|2+|u|2)dx
1
(cid:18)ZΩ1 (cid:19)
and
1
2
||u|| = (|∇u|2+|u|2)dx
2
(cid:18)ZΩ2 (cid:19)
respectively.
In order to show that the set M is not empty, we need of the following Lemma.
Lemma 2.1. Let v ∈ H1(Ω) with v 6= 0 for j = 1,2. Then, there are t,s > 0 such that
0 j
J′(tv +sv )v = 0 and J′(tv +sv )v = 0.
1 2 1 1 2 2
Proof. It what follows, we consider the vector field
H(s,t)= J′(tv +sv )(tv ),J′(tv +sv )(sv ) .
1 2 1 1 2 2
(cid:0) (cid:1)
From (f )−(f ), a straightforward computation yields that there are 0 < r < R such that
1 3
J′(rv +sv )(rv ), J′(tv +rv )(rv ) > 0, ∀s,t∈ [r,R]
1 2 1 1 2 2
and
J′(Rv +sv )(Rv ), J′(tv +Rv )(Rv )< 0, ∀s,t∈ [r,R].
1 2 1 1 2 2
Now, the lemma follows by applying Miranda theorem [23].
As an immediate consequence of the last lemma, we have the following corollary
Corollary 2.2. The set M is not empty.
Next, we will show some technical lemmas.
Lemma 2.3. There exists ρ > 0 such that
(i) J(u) ≥ ||u||2/4 and ||u|| ≥ ρ,∀u∈ N;
(ii) ||w || ≥ ρ, ∀w ∈M and j = 1,2, where w = w| ,j = 1,2.
j j j Ωj
Proof. From (f ), for any u∈ N
4
4J(u) = 4J(u)−J′(u)u = ||u||2 + [uf(u)−4F(u)]dx ≥ ||u||2
ZΩ
7
and so,
J(u) ≥ ||u||2/4, ∀u∈ N.
From (f ) and (f ), there is C > 0 such that
1 2
λ
f(s)s≤ 1s2+Cs6, for all s ∈R,
2
where λ is the first eigenvalue of (−∆,H1(Ω)). Since J′(u)u = 0,
1 0
λ
||u||2 < ||u||2+ φ u2dx = uf(u)dx ≤ 1 u2dx+C u6dx.
u
2
ZΩ ZΩ ZΩ ZΩ
Then, by Sobolev embeddings,
1
||u||2 < ||u||2+Cˆ||u||6,
2
from where it follows that
||u|| ≥ ρ ∀u∈ N,
1
where ρ= 1 4, finishing the proof of (i).
2Cˆ
If w ∈ M(cid:16) , (cid:17)we have that J′(w)w = J′(w)w = 0. Then, a simple computation gives
1 2
J′(w )w < 0 for j = 1,2, which implies
j j
||w ||2 < ||w ||2+ φ (w )2dx < f(w )w dx, for j = 1,2.
j j j j wj j j j
ZΩj ZΩj
As in (i), we can deduce that ||w || ≥ ρ for j = 1,2.
j j
Lemma 2.4. If (w ) is a bounded sequence in M and p ∈ (2,6), we have
n
liminf |w |pdx > 0 j = 1,2.
n,j
n
ZΩj
where w = w | for j = 1,2.
n,j n Ωj
Proof. From (f ) and (f ), given ε> 0 there exists C > 0 such that
1 2
f(s)s≤ ελ s2+C|s|p+εs6, for all s ∈ R.
1
Since w ∈ M, by Lemma 2.3
n
ρ2 ≤ ||w ||2 < w f(w )dx ≤ ελ (w )2dx+C |w |pdx+ε (w )6dx
n,j j n,j n,j 1 n,j n,j n,j
ZΩj ZΩj ZΩj ZΩj
that is,
ρ2 ≤ ε λ (w )2dx+ (w )6dx +C |w |pdx.
1 n,j n,j n,j
ZΩj ZΩj ! ZΩj
8
Using the boundedness of (w ), there is C such that
n 1
ρ2 ≤ εC +C |w |pdx.
1 n,j
ZΩj
Fixing ε = ρ2 , we get
2C1
ρ2
|w |pdx ≥ ,
n,j
2C
ZΩj
showing that
ρ2
liminf |w |pdx ≥ > 0.
n,j
n 2C
ZΩj
2.2 Existence of least energy solution for (P)
∞,Υ
In this subsection, our main goal is to prove the following result
Theorem 2.5. Assume that (f )−(f ) hold. Then equation (P) possesses a positive
1 4 ∞,Υ
least energy solution on the set M.
Proof. In what follows, we denote by c the infimum of J on M, that is,
0
c = inf J(v).
0
v∈M
From Lemma 2.3(i), we conclude that c > 0.
0
By Corollary 2.2, we know that M is not empty, then there is a sequence (w ) ⊂ M
n
satisfying
limJ(w ) = c .
n 0
n
Still from Lemma 2.3(i), (w ) is a bounded sequence. Hence, without loss of generality,
n
we may suppose that there is w ∈H1(Ω) verifying
0
w ⇀ w in H1(Ω),
n 0
w → w in Lp(Ω) ∀p∈ [1,2∗)
n
and
w (x) → w(x) a.e. in Ω.
n
Then, (f ) combined with the compactness lemma of Strauss [7, Theorem A.I, p.338] gives
2
lim |w |pdx = |w |pdx,
n,j j
n
ZΩj ZΩj
lim w f(w )dx = w f(w )dx
n,j n,j j j
n
ZΩj ZΩj
9
and
lim F(w )dx = F(w )dx,
n,j j
n
ZΩj ZΩj
from where it follows together with Lemma 2.4 that w 6= 0 for j = 1,2. Then, by Lemma
j
2.1 there are t,s > 0 verifying
J′(tw +sw )w = 0 and J′(tw +sw )w = 0.
1 2 1 1 2 2
Next, we will show that t,s ≤ 1. Since J′(w )w = 0 for j = 1,2,
n,j n,j
||w ||2 + φ (w )2dx+ φ (w )2dx = f(w )w dx
n,1 1 wn,1 n,1 wn,2 n,1 n,1 n,1
ZΩ1 ZΩ1 ZΩ1
and
||w ||2+ φ (w )2dx+ φ (w )2dx = f(w )w dx.
n,2 2 wn,2 n,2 wn,1 n,2 n,2 n,2
ZΩ2 ZΩ2 ZΩ2
Taking the limit in the above equalities, we obtain
||w ||2+ φ (w )2dx+ φ (w )2dx ≤ f(w )w dx
1 1 w1 1 w2 1 1 1
ZΩ1 ZΩ1 ZΩ
and
||w ||2+ φ (w )2dx+ φ (w )2dx ≤ f(w )w dx.
2 2 w2 2 w1 2 2 2
ZΩ2 ZΩ2 ZΩ2
Recalling that
J′(tw +sw )(tw )= J′(tw +sw )(sw )= 0,
1 2 1 1 2 2
it follows that
t2||w ||2+t4 φ (w )2dx+t2s2 φ (w )2dx = f(tw )tw dx
1 2 w1 1 w2 1 1 1
ZΩ1 ZΩ1 ZΩ1
and
s2||w ||2+s4 φ (w )2dx+t2s2 φ (w )2dx = f(sw )sw dx.
2 2 w2 2 w2 1 2 2
ZΩ2 ZΩ2 ZΩ2
Now, without loss of generality, we will suppose that s ≥ t. Under this condition,
s2||w ||2 +s4 φ (w )2dx+s4 φ (w )2dx ≥ f(sw )sw dx
2 2 w2 2 w2 1 2 2
ZΩ2 ZΩ2 ZΩ2
and then
1 f(sw )sw f(w )w
−1 ||w ||2 ≥ 2 2 − 2 2 (w )4dx.
s2 2 2 (sw )4 (w )4 2
(cid:18) (cid:19) ZΩ2(cid:18) 2 2 (cid:19)
If s> 1, the left side in this inequality is negative, but from (f ), the right side is positive,
4
thus we must have s ≤ 1, which also implies that t ≤ 1.
Our next step is to show that J(tw +sw ) = c . Recalling that tw +sw ∈ M, we
1 2 0 1 2
derive that
1
c ≤ J(tw +sw ) = J(tw +sw )− J′(tw +sw )(tw +sw ).
0 1 2 1 2 1 2 1 2
4
10
