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University of Mazandaran, Iran http://cjms.journals.umz.ac.ir ISSN: 1735-0611 CJMS

.

4(1)(2015), 61-75

Existence of infinitely many solutions for coupled system

of Schr¨

o

dinger-Maxwell’s equations

G.Karamali

1

and M.K.Kalleji

2

1

PhD, Assistant Professor, Academic member of School of

Mathematics, Shahid Sattari Aeronautical University of Science and

Technology, South Mehrabad, Tehran, Iran.

2

School of Mathematics, Shahid Sattari Aeronautical University of

Science and Technology, South Mehrabad, Tehran, Iran.

Abstract. In this paper we study the existence of infinitely many large energy solutions for the coupled system of Schr¨odinger-Maxwell’s equations

     

    

−∆u+V(x)u+φu=Hv(x, u, v) inR3 −∆φ=u2 inR3

−∆v+V(x)v+ψv=Hu(x, u, v) inR3

−∆ψ=v2 in

R3,

via the Fountain theorem under Cerami condition. More precisely, we consider the More general case and weakenV1∗with respect to

the conditionV1 in [6].

Keywords: Schr¨odingerMaxwell system ,Cerami condition,Variational methods, Strongly indefinite functionals.

2000 Mathematics subject classification: 35Pxx; Secondary 46Txx.

1Corresponding author: m.kalleji@yahoo.com

Received: 30 November 2014 Revised: 15 March 2015 Accepted: 16 March 2015

(2)

1.

Introduction

In this paper, we study the nonlinear coupled system of Schr¨

o

dinger-Maxwell’s equations

u

+

V

(

x

)

u

+

φu

=

H

v

(

x, u, v

)

in

R

3

φ

=

u

2

in

R

3

v

+

V

(

x

)

v

+

ψv

=

Hu

(

x, u, v

)

in

R

3

ψ

=

v

2

in

R

3

,

(1.1)

where

V

C

(

R

3

,

R

) and

H

C

1

(

R

3

,

R

) which are satisfied in some

suitable conditions.

In the classical model, the interaction of a charge particle with an

electro-magnetic field can be described by the nonlinear Schr¨

o

dinger-Maxwell’s

equations.

In this article, we want to study the interaction of two

charge particles Simultaneously with same potential function

V

(

x

) and

different scalar potential

φ

and

ψ

which are satisfied in suitable

condi-tions. For more details on the physical aspects see [1] and the references

therein. More precisely, we have to solve the system 1.1 if we want to

find electrostatic-type solutions.

In [2] Zhang et al. considered the system 1.1 without the scalar potential

φ

and

ψ

, so-called the Hamiltonian elliptic system

u

+

V

(

x

)

u

=

H

v

(

x, u, v

)

in

R

3

v

+

V

(

x

)

v

=

H

u

(

x, u, v

)

in

R

3

,

(1.2)

They assumed the potential

V

(

x

) is non-periodic and sing changing,

and

H

(

x, z

) is non-periodic and asymptotically quadratic in

z

= (

u, v

) :

R

N

R

×

R

.

For the case of a bounded domain the system 1.2 were

studied by many authors [3, 4, 5] and the references therein.

In [6] Li and Chen, studied the existence of infinitely many large energy

solutions for the superlinear Schr¨

o

dinger-Maxwell’equations

u

+

V

(

x

)

u

+

φu

=

f

(

x, u

)

in

R

3

φ

=

u

2

in

R

3

,

(1.3)

They assumed on this paper that the potential function

V

(

x

) is bounded

from below with a positive constant and they are used to solve the

prob-lem 1.3, the Fountain theorem in critical point theory. In [7] Schaftingen

et al. studied positive bound states for the equation

2

u

+

V

(

x

)

u

=

K

(

x

)

f

(

u

)

in

R

N

,

(1.4)

(3)

as nonlinear parametric Schr¨

o

dinger equation which were studied by

many authors see [8, 9, 10]. The infinitely many large solutions for the

equation 1.3 are obtained in [11] with the following variant

Ambrosetti-Rabinowitz type condition [6], there exist

µ >

4 such that for any

s

R

and

x

R

3

µF

(

x, u

) :=

µ

Z

s

0

f

(

x, t

)

dt

sf

(

x, s

)

.

Existence of solutions are obtained via Fountain theorem in critical point

theory. More precisely, in this paper we consider the more general case

and weaken the condition of

V

1

in [6] and we assume that the potential

V

is non-periodic and sing changing. We assume the following conditions

:

V

1

)

V

C

(

R

3

,

R

) and there exists some

M >

0 such that the set

M

=

{

x

R

3

;

V

(

x

)

M

}

is not nonempty and has finite Lebesgue

measure.

H

1

)

H

C

1

(

R

3

×

R

2

,

R

) and for some 2

< p <

2

= 6

,

and

M

1

, M

2

>

0

,

|

Hu

(

x, u, v

)

| ≤

M

1

|

u

|

+

M

1

|

u

|

p−1

and

|

Hv

(

x, u, v

)

| ≤

M

2

|

v

|

+

M

2

|

v

|

p−1

,

for a.e

x

R

3

and

u, v

:

R

3

R

,

and also

lim

u→0

Hu

(

x, u, v

)

u

= 0

and

u

lim

→0

Hv

(

x, u, v

)

v

= 0

,

uniformly for

x

R

3

and

u, v

R

.

H

2

)

lim

|(u,v)|→∞

H(x,u,v)

|(u,v)|4

= +

,

uniformly in

x

R

3

and (

u, v

)

R

2

and

H

(

x,

0

,

0) = 0

, H

(

x, u, v

)

0

for all (

x, u, v

)

R

3

×

R

×

R

.

H

3

) There exists a constant

θ

1 such that

θ

H

ˆ

(

x, u, v

)

H

ˆ

(

x, su, sv

)

for all

x

R

3

,

(

u, v

)

R

2

and

t, s

[0

,

1]

,

where

ˆ

H

(

x, u, v

) =

Hu

(

x, tu, v

)

tu

+

Hv

(

x, u, sv

)

sv

4

H

(

x, tu, sv

)

.

H

4

)

H

(

x,

u, v

) =

H

(

x, u, v

) and

H

(

x, u,

v

) =

H

(

x, u, v

) for all

x

R

3

and (

u, v

)

R

2

.

Here, we express the Cerami condition which was established by G.

Cerami in [12]

Definition 1.1.

Suppose that functional

I

is

C

1

and

c

R

,

if any

sequence

{

u

n

}

satisfying

I

(

u

n

)

c

and (1 +

k

u

n

k

)

I

0

(4)

To approach the main result, we need the following critical point

theorem.

Theorem 1.2.

(Fountain theorem under Cerami condition) Let

X

be a

Banach space with the norm

k

.

k

and let

X

j

be a sequence of subspace of

X

with

dimX

j

<

for any

j

N

.

Further,

X

=

j∈N

Xj,

the closure

of the direct sum of all

X

j

.

Set

W

k

=

kj=0

X

j

, Z

k

∞j=k

X

j

.

Consider

an even functional

I

C

1

(

X,

R

)

,

that is ,

I

(

u

) =

I

(

u

)

for any

u

X.

Also suppose that for any

k

N

,

there exist

ρ

k

> r

k

>

0

such that

I

1

)

ak

:= max

u∈Wk,kuk=ρk

I

(

u

)

0

,

I

2

)

bk

:= inf

u∈Zk,kuk=rk

I

(

u

)

+

as

k

→ ∞

,

I

3

)

the Cerami condition holds at any level

c >

0

.

Then the functional

I

has an unbounded sequence of critical values.

Now, our main result is the following :

Theorem 1.3.

Let

V

1

, H

1

H

4

be satisfied. Then the system 1.1 has

infinitely many solutions

{

((

u

k

, φ

k

)

,

(

v

k

, ψ

k

))

}

in product space

YHD

×

Y

HD

(see section 2)which satisfies in

1

2

Z

R3

[

|∇

u

k

|

2

+

|∇

v

k

|

2

+

V

(

x

)(

u

2k

+

v

k2

) ]

dx

1

4

Z

R3

[

|∇

φ

k

|

2

+

|∇

ψ

k

|

2

]

dx

+

1

2

Z

R3

[

φ

k

u

2k

+

ψ

k

v

k2

]

dx

Z

R3

H

(

x, u, v

)

dx

+

.

Remark

1.4

.

The assumption

V

1

implies that the potential

V

is not

periodic and changes sing. This is different from condition

V

1

in [6].

Also conditions

H

1

and

H

3

modified for the system 1.1 with respect to

the system 1

.

1 which considered by Li and Chen in [6].

2.

some auxiliary results and notations

In this section we give some notations and definitions on the function

product space. We set

H

1

(

R

3

) :=

{

u

L

2

(

R

3

)

| |∇

u

| ∈

L

2

(

R

3

)

}

,

(2.1)

with the norm

k

u

k

H1

:= (

Z

R3

|∇

u

|

2

+

u

2

dx

)

12

,

(2.2)

and we consider the function space

D

1,2

(

R

3

) :=

{

u

L

2

(5)

with the norm

k

u

k

D1,2

:= (

Z

R3

|∇

u

|

2

dx

)

12

.

(2.4)

Now, we consider the function space

E

:=

{

u

H

1

(

R

3

)

|

Z

R3

|∇

u

|

2

+

u

2

dx <

∞}

.

Then

E

is Hilbert space [13] with the inner product

(

u, v

)

E

:=

Z

R3

(

u.

v

+

V

(

x

)

uv

)

dx

(2.5)

and the norm

k

u

k

E

:= (

u, u

)

1 2

E

.

We set

X

E

:=

E

×

E, Y

HD

:=

H

1

(

R

3

)

×

D

1,2

(

R

3

)

and Z

ED

:=

E

×

D

1,2

(

R

3

)

.

Hence , we can define an inner product on

XE

as

((

u, v

)(

w, z

))

XE

:= (

u, w

)

E

+ (

v, z

)

E

(2.6)

and the corresponding norm on

X

E

by this inner product as following

k

(

u, v

)

k

XE

:= (

k

u

k

2

E

+

k

v

k

2E

)

1

2

= ((

u, u

)

E

+ (

v, v

)

E

)

1

2

.

(2.7)

Let

H

be a Hilbert space and

H

=

H

×

H

with the inner product

((

u, v

)

,

(

w, z

))

H

= (

u, w

)

H

+ (

v, z

)

H

and the corresponding generated norm by this inner product. Suppose

that

T

is operator on Hilbert space

H

. We assume that there is an

orthogonal decomposition

H

=

H

H

+

H

0

such that

T

is negative definite (resp. positive definite) in

H

(

resp.H

+

)

and

H

0

=

kerT.

Let

H

+

=

H

+

×

H

,

H

=

H

×

H

+

and

H

0

=

H

0

×

H

0

.

Then for any

z

= (

u, v

)

∈ H

we have

z

=

z

+

z

+

+

z

0

,

where

z

+

= (

u

+

, v

)

, z

= (

u

, v

+

) and

z

0

= (

u

0

, v

0

)

.

Therefore,

H

+

,

H

and

H

0

are orthogonal. Hence,

H

=

H

+

⊕ H

⊕ H

0

,

for details see [2].

Lemma 2.1.

[2]

If

V

1

holds. Then

H

,

L

p

(

R

N

,

R

2

)

is continuous for

p

[2

,

2

]

and

H

,

L

ploc

(

R

N

,

R

2

)

is compact for

p

[2

,

2

)

.

Therefore, the system 1.1 is the Euler-Lagrange equations of the

func-tional

(6)

define by

J

((

u, φ

)

,

(

v, ψ

)) :=

1

2

k

(

u, v

)

k

2 XE

1

4

Z

R3

[

|∇

φ

|

2

+

|∇

ψ

|

2

]

dx

+

1

2

Z

R3

[

φu

2

+

ψv

2

]

dx

Z

R3

H

(

x, u, v

)

dx.

(2.8)

The functional

J

C

1

(

Z

ED

×

Z

ED

,

R

) and its critical points are the

solutions of system 1.1.

Remark

2.2

.

[6] the functional

J

exhibits a strong indefiniteness, that

is, it is unbounded both from below and from above on infinitely

dimen-sional subspaces. This indefiniteness can be removed using the reduction

method described in [14], by which we are led to study a one variable

functional that does not present such a strongly indefiniteness nature.

For any

u

E

the Lax-Milgram theorem [15] implies there exists a

unique

φu

D

1,2

(

R

3

) such that

φ

u

=

u

2

in a distributional (or weak) sense. Then we can obtain an integral

expression for

φu

:

φ

u

=

1

4

π

Z

R3

u

2

(

y

)

|

x

y

|

dy,

(2.9)

for any

u

E.

Lemma 2.3.

[11]

For any

u

E

i

)

k

φu

k

D1,2

M

3

k

u

k

2

L125

,

where

M

3

is positive constant which does

not depend on

u.

In particular, there exists positive constant

M

4

such

that

Z

R3

φ

u

u

2

dx

M

4

k

u

k

4E

;

(2.10)

ii

)

φu

0

.

Now, by the lemma 2.3 we define the functional

I

:

X

E

R

by

I

(

u, v

) :=

J

((

u, φ

u

)

,

(

v, ψ

v

))

.

Remark

2.4

.

Using the relation

φ

u

=

u

2

and integration by parts,

we can obtain

Z

R3

|∇

φ

u

|

2

dx

=

Z

R3

(7)

Then, we can consider the functional 2.8 as following

I

(

u, v

) =

1

2

k

(

u, v

)

k

2 XE

+

1

4

Z

R3

[

φuu

2

+

ψvv

2

]

dx

Z

R3

H

(

x, u, v

)

dx.

(2.11)

It is well-known that

I

is

C

1

-functional with derivative given by

h

I

0

(

u, v

)

,

(

u, v

)

i

=

Z

R3

[

u.

u

+

V

(

x

)

u

2

+

φ

u

u

2

H

u

(

x, u, v

) ]

dx

+

Z

R3

[

v.

v

+

V

(

x

)

v

2

+

ψ

v

v

2

H

v

(

x, u, v

) ]

dx.

(2.12)

Now, using the proposition 2.3 in [6] we can consider the following

proposition for our functional

J

:

Proposition 2.5.

The following statements are equivalent :

i

) ((

u, φu

)

,

(

v, ψv

))

Z

ED

×

Z

ED

is a critical point of

J

;

ii

) (

u, v

)

is a critical point of functional

I

and

(

φ, ψ

) = (

φ

u

, ψ

v

)

.

Proof.

It follows using the remark 2.2 and theorem 2.3 in [14].

3.

proof of main theorem

We take an orthogonal basis

{

(

e

i

, e

j

)

}

of product space

X

:=

X

E

=

E

×

E

and we define

W

k

:=

span

{

(

e

i

, e

j

)

}

i,j=1,...,k

, Z

k

:=

W

k⊥

.

Lemma 3.1.

[11]

for any

p

[2

,

2

)

,

β

k

:=

sup

u∈Zk,kuk=1

k

u

k

Lp

0

as

k

→ ∞

.

Now, we prove that the functional

I

:

X

E

R

satisfies the Cerami

condition .

Proposition 3.2.

under the conditions

H

1

H

3

,

the functional

I

(

u, v

)

satisfies the Cerami condition at any positive level.

Proof.

Let

{

(

u

n

, v

n

)

}

be a sequence in

X

E

such that for some

c

R

,

I

(

un, vn

) =

1

2

k

(

un, vn

)

k

2 XE

+

1

4

Z

R3

[

φu

n

u

2 n

+

ψv

n

v

2 n

]

dx

Z

R3

H

(

x, un, vn

)

dx

c

(3.1)

and

(1 +

k

(

un, vn

)

k

XE

)

I

0

(

un, vn

)

0

, as n

→ ∞

.

(3.2)

From 3.1 and 3.2 for n large enough

1 +

c

I

(

un, vn

)

1

4

h

I

0

(8)

1

2

k

(

u

n

, v

n

)

k

2 XE

+

1

4

Z

R3

[

φ

un

u

2

n

+

ψ

vn

v

2 n

]

dx

Z

R3

H

(

x, u

n

, v

n

)

dx

1

4

[

Z

R3

[

un.

un

+

V

(

x

)

u

2n

+

φu

n

u

2

n

Hu

n

(

x, un, vn

)

un

]

dx

+

Z

R3

[

vn.

vn

+

V

(

x

)

v

n2

+

ψv

n

v

2

n

Hv

n

(

x, un, vn

)

vn

]

dx

]

1

2

k

(

u

n

, v

n

)

k

2 XE

Z

R3

H

(

x, u

n

, v

n

)

dx

1

4

[

Z

R3

[

u

n

.

u

n

+

V

(

x

)

u

2n

]

dx

+

Z

R3

[

v

n

.

v

n

+

V

(

x

)

v

n2

]

dx

]

+

1

4

[

Z

R3

Hu

n

(

x, un, vn

)

undx

+

Z

R3

Hv

n

(

x, un, vn

)

vndx

] =

1

4

k

(

un, vn

)

k

2 XE

+

1

4

[

Z

R3

Hu

n

(

x, un, vn

)

undx

+

Z

R3

Hv

n

(

x, un, vn

)

vndx

]

Z

R3

H

(

x, un, vn

)

dx.

(3.3)

Now, we shall show that the sequence

{

(

u

n

, v

n

)

}

is bounded. Suppose

that

k

(

un, vn

)

k

XE

→ ∞

as

n

→ ∞

.

Then we consider

(

w

n

, z

n

) := (

u

n

k

u

n

k

,

v

n

k

v

n

k

)

X

E

,

so the sequence

{

(

wn, zn

)

}

is bounded.

For some (

w, z

)

XE,

and

subsequence of (

w

n

, z

n

)

,

we can imply that

(

wn, zn

)

*

(

w, z

)

as n

→ ∞

in XE,

and

(

w

n

, z

n

)

(

w, z

)

in L

t

(

R

3

)

×

L

s

(

R

3

)

f or t, s

[2

,

2

)

and

w

n

(

x

)

w

(

x

)

and z

n

(

x

)

z

(

x

)

(3.4)

for a.e.

x

R

3

.

Now, we consider two cases. In first case we suppose

that (

w, z

)

6

= (0

,

0) in

X

E

.

By dividing relation 3.1 with

k

(

un, vn

)

k

4XE

and lemma 2.3 we get that

Z

R3

H

(

x, u

n

, v

n

)

k

(

un, vn

)

k

4 XE

dx

=

1

2

k

(

un, vn

)

k

2 XE

+

Z

R3

(

φ

un

u

2

n

+

ψ

vn

v

2

n

)

dx

c

4

k

(

un, vn

)

k

4

XE

+

O

(

k

(

un, vn

)

k

−4X

E

)

M

5

<

,

(3.5)

where

M

5

is a positive constant. we consider Ω :=

{

x

R

3

|

w

(

x

)

, z

(

x

)

6

=

0

}

.

By condition

H

2

for all

x

,

H

(

x, u

n

, v

n

)

k

(

un, vn

)

k

4 XE

=

H

(

x, u

n

, v

n

)

|

(

un, vn

)

|

4

(

w

4

(9)

Since

|

|

>

0

,

by Fatou’s lemma,

Z

R3

H

(

x, un, vn

)

k

(

u

n

, v

n

)

k

4XE

dx

+

as n

→ ∞

.

This contradicts with 3.5. In second case, we assume that (

wnzn

) =

(0

,

0) then we define sequence (

t

n

, s

n

)

R

2

by

I

(

tnun, snvn

)

max

(t,s)∈[0,1]×[0,1]

I

(

tu, sv

)

.

Let

m >

0 be fixed and let

( ¯

w,

z

¯

) := (

4

m

un

k

un

k

E

,

4

m

un

k

un

k

E

) = 2

m

(

wn, zn

)

XE

.

By condition

H

1

we have

|

Hu

(

x, u, v

)

| ≤

M

1

|

u

|

2

+

M

1

|

u

|

p−1

and

|

Hv

(

x, u, v

)

| ≤

M

2

|

v

|

2

+

M

2

|

u

|

p−1

for any

x

R

3

and (

u, v

)

R

2

,

and

H

(

x, u, v

) =

Z

1

0

Hu

(

x, tu, v

)

tudt

+

Z

1

0

Hv

(

x, u, sv

)

svds.

Therefore,

H

(

x, u, v

)

M

1

2

|

u

|

2

+

M

1

p

|

u

|

p

+

M

2

2

|

v

|

2

+

M

2

p

|

v

|

p

.

(3.6)

Using the relation 3.5 we get that

lim

n→∞

Z

R3

H

(

x,

w

¯

n

,

z

¯

n

)

dx

lim

n→∞

[

Z

R3

M

1

2

|

w

¯

n

|

2

+

M

1

p

|

w

¯

n

|

p

+

M

2

2

|

z

¯

n

|

2

+

M

2

p

|

z

¯

n

|

p

]

dx

lim

n→∞

M

6

Z

R3

[

|

w

¯

n

|

2

+

|

z

¯

n

|

2

]

dx

+

M

7

Z

R3

[

|

w

¯

n

|

p

+

|

z

¯

n

|

p

]

dx,

where

M

6

:= max

{

M22

,

M23

}

and

M

7

:= max

{

Mp2

,

Mp3

}

.

Hence,

lim

n→∞

Z

R3

H

(

x,

w

¯

n

,

z

¯

n

)

dx

lim

n→∞

M

6

Z

R3

[

|

w

¯

n

|

2

+

|

z

¯

n

|

2

]

dx

+ lim

n→∞

M

7

Z

R3

[ (

|

w

¯

n

|

2

+

|

z

¯

n

|

2

)

p

]

dx

lim

n→∞

M

6

Z

R3

|

( ¯

wn,

zn

¯

)

|

2

dx

+

M

7

Z

R3

(

|

( ¯

wn,

zn

¯

)

|

2

)

p

dx

= 0

.

(3.7)

Therefore, for

n

large enough,

I

((

t

n

u

n

, s

n

v

n

))

I

(( ¯

w

n

,

z

¯

n

)) = 2

m

+

1

4

Z

R3

[

φ

w¯n

w

¯

2

n

+

ψ

z¯n

z

¯

2 n

]

dx

Z

R3

(10)

By relation 3.8, lim

n→∞

I

(

tnun, snvn

) = +

.

Since

I

(0

,

0) = 0 and

I

(

un, vn

)

c

then

t

n

s

n

(0

,

1)

.

Hence, for

t

n

, s

n

(0

,

1) and

n

large enough we

ob-tain

Z

R3

t

n

u

n

.

t

n

u

n

+

V

(

x

)

t

n

u

n

t

n

u

n

+

φ

tnun

t

n

u

n

t

n

u

n

H

tnun

(

x, t

n

u

n

, v

n

)

t

n

u

n

dx

+

Z

R3

snvn.

snvn

+

V

(

x

)

snvnsnvn

+

φs

nvn

snvnsnvn

Hs

nvn

(

x, un, snvn

)

snvndx

=

h

I

0

((

tnun, snvn

))

,

(

tnun, snvn

)

i

= 0

.

Then by

H

3

we can get that

I

((

un, vn

))

1

4

h

I

0

(

un, vn

)

,

(

un, vn

)

i

=

1

4

k

(

u

n

, v

n

)

k

2 XE

+

1

4

Z

R3

[

H

un

(

x, u

n

, v

n

)

u

n

+

H

vn

(

x, u

n

, v

n

)]

dx

Z

R3

H

(

x, u

n

, v

n

)

dx

=

1

4

k

(

u

n

, v

n

)

k

2 XE

+

1

4

Z

R3

ˆ

H

(

x, u

n

, v

n

)

dx

1

4

θ

k

(

u

n

, v

n

)

k

2 XE

+

1

4

θ

Z

R3

[

H

tnun

(

x, t

n

u

n

, v

n

)

t

n

u

n

+

H

snvn

(

x, u

n

, s

n

v

n

)

s

n

v

n

]

dx

Z

R3

H

(

x, t

n

u

n

, s

n

v

n

)

dx

1

θ

I

((

tnun, snvn

))

1

4

θ

h

I

0

((

tnun, snvn

))

,

(

tnun, snvn

)

i →

+

.

This is contradiction with 3.3. Hence, the sequence

{

(

u

n

, v

n

)

}

is bounded

in

XE.

Since (

un, vn

)

,

(

u, v

) in

XE

,

so by lemma 2.1, (

un, vn

)

(

u, v

)

in for any

s, t

[2

,

2

)

.

By 2.12 we can get that

k

(

u

n

, v

n

)

(

u, v

)

k

X2E

=

k

(

u

n

u, v

n

v

)

k

2

XE

= ((

u

n

u, v

n

v

)

,

(

u

n

u, v

n

v

))

XE

=

Z

R3

(

u

n

u

)

.

(

u

n

u

)

V

(

x

)(

u

n

u

)

2

dx

+

Z

R3

(

v

n

v

)

.

(

v

n

v

)

V

(

x

)(

v

n

v

)

2

dx

=

h

I

0

(

u

n

, v

n

)

I

0

(

u, v

)

,

(

u

n

, v

n

)

(

u, v

)

i −

Z

R3

[ (

φ

un

u

n

φ

u

u

)(

u

n

u

) ]

dx

+

Z

R3

[ (

ψ

vn

v

n

ψ

v

v

)(

v

n

v

) ]

dx

+

Z

R3

[ (

H

un

(

x, u

n

, v

n

)

H

u

(

x, u, v

))(

u

n

u

) ]

dx

+

Z

R3

(11)

Since (

un, vn

)

,

(

u, v

) in

X

E

,

and (

un, vn

)

(

u, v

) in for any

s, t

[2

,

2

)

,

we can imply

h

I

0

(

u

n

, v

n

)

I

0

(

u, v

)

,

(

u

n

, v

n

)

(

u, v

)

i →

0

,

as

n

→ ∞

.

On the other hand,

|

Z

R3

[ (

φu

n

un

φuu

)(

un

u

) ]

dx

+

Z

R3

[ (

ψv

n

vn

ψv

v

)(

vn

v

) ]

dx

| ≤

|

Z

R3

[ (

φu

n

un

φuu

)(

un

u

) ]

dx

|

+

|

Z

R3

[ (

ψv

n

vn

ψvv

)(

vn

v

) ]

dx

|

.

By H¨

o

der inequality and 2.3,

|

φ

un

u

n

(

u

n

u

)

dx

| ≤ k

φ

un

u

n

k

L2

k

u

n

u

k

L2

≤ k

φ

un

k

L6

k

u

n

k

L3

k

u

n

u

k

L2

M

8

k

φu

n

k

D1,2

k

un

k

L3

k

un

u

k

L2

M

3

M

8

k

un

k

2

L125

k

un

k

L

3

k

un

u

k

L2

,

(3.9)

where

M

8

is positive constant. We can similarity conclude the following

inequalities :

|

φuu

(

un

u

)

dx

| ≤ k

φuu

k

L2

k

un

u

k

L2

≤ k

φu

k

L6

k

un

k

L3

k

un

u

k

L2

M

9

k

φ

u

k

D1,2

k

u

n

k

L3

k

u

n

u

k

L2

M

3

M

9

k

u

k

2 L125

k

u

k

L3

k

u

n

u

k

L2

,

(3.10)

and

|

ψ

vn

v

n

(

v

n

v

)

dx

| ≤ k

ψ

vn

v

n

k

L2

k

v

n

v

k

L2

≤ k

ψ

vn

k

L6

k

v

n

k

L3

k

v

n

v

k

L2

M

10

k

ψv

n

k

D1,2

k

vn

k

L3

k

vn

v

k

L2

M

3

M

10

k

vn

k

2L125

k

vn

k

L3

k

vn

v

k

L2

,

(3.11)

and

|

ψ

v

v

(

v

n

v

)

dx

| ≤ k

ψ

v

v

k

L2

k

v

n

v

k

L2

≤ k

ψ

v

k

L6

k

v

n

k

L3

k

v

n

v

k

L2

M

11

k

ψ

v

k

D1,2

k

v

n

k

L3

k

v

n

v

k

L2

M

3

M

11

k

v

k

2

L125

k

v

k

L

3

k

v

n

v

k

L2

,

(3.12)

where

M

9

, M

10

and

M

11

are positive constants.

Therefore, by relations 3.9, 3.10, 3.11 and 3.12 we obtain that

|

Z

R3

[ (

φ

un

u

n

φ

u

u

)(

u

n

u

) ]

dx

+

Z

R3

[ (

ψ

vn

v

n

ψ

v

v

)(

v

n

v

) ]

dx

| ≤

[

M

3

M

8

k

un

k

2

L125

k

un

k

L

3

+

M

3

M

9

k

u

k

2

L125

k

u

k

L

3

]

k

un

u

k

L2

+[

M

3

M

10

k

v

n

k

2L125

k

v

n

k

L3

+

M

3

M

11

k

v

k

2

L125

k

v

k

L

3

]

k

v

n

v

k

L2

.

Then

Z

R3

[ (

φ

un

u

n

φ

u

u

)(

u

n

u

) ]

dx

+

Z

R3

(12)

Now, by condition

H

1

and H¨

o

der inequality we obtain that

|

Z

R3

H

un

(

x, u

n

, v

n

)(

u

n

u

)

dx

| ≤

Z

R3

|

H

un

(

x, u

n

, v

n

)

| |

u

n

u

|

dx

Z

R3

[

M

1

2

|

u

n

|

+

M

1

p

|

u

n

|

p−1

]

|

u

n

u

|

dx

[

M

1

2

k

un

k

2 L2

+

M

1

p

k

un

k

p−1

LP

]

k

un

u

k

LP

.

(3.13)

Similarly, we can get the following inequalities by condition

H

1

and

o

der inequality:

|

Z

R3

Hu

(

x, u, v

)(

un

u

)

dx

| ≤

Z

R3

|

Hu

(

x, u, v

)

| |

un

u

|

dx

Z

R3

[

M

1

2

|

u

|

+

M

1

p

|

u

|

p−1

]

|

un

u

|

dx

[

M

1

2

k

u

k

2 L2

+

M

1

p

k

u

k

p−1

LP

]

k

u

n

u

k

LP

(3.14)

and

|

Z

R3

Hv

n

(

x, un, vn

)(

vn

v

)

dx

| ≤

Z

R3

|

Hv

n

(

x, un, vn

)

| |

vn

v

|

dx

Z

R3

[

M

2

2

|

vn

|

+

M

2

p

|

vn

|

p−1

]

|

vn

v

|

dx

[

M

2

2

k

v

n

k

2 L2

+

M

2

p

k

v

n

k

p−1

LP

]

k

v

n

v

k

LP

(3.15)

and

|

Z

R3

H

v

(

x, u, v

)(

v

n

v

)

dx

| ≤

Z

R3

|

H

v

(

x, u, v

)

| |

v

n

v

|

dx

Z

R3

[

M

2

2

|

v

|

+

M

2

p

|

v

|

p−1

]

|

vn

v

|

dx

[

M

2

2

k

v

k

2 L2

+

M

2

p

k

v

k

p−1

LP

]

k

v

n

v

k

LP

.

(3.16)

Hence, by relations 3.13, 3.14, 3.15 and 3.16 we obtain

Z

R3

[ (

H

un

(

x, u

n

, v

n

)

H

u

(

x, u, v

))(

u

n

u

) ]

dx

+

Z

R3

(13)

2[

M

1

2

k

u

n

k

2 L2

+

M

1

p

k

u

n

k

p−1

LP

]

k

u

n

u

k

LP

+2[

M

2

2

k

v

n

k

2 L2

+

M

2

p

k

v

n

k

p−1

LP

]

k

v

n

v

k

LP

.

Since, (

u

n

, v

n

)

(

u, v

) in

L

t

(

R

3

)

×

L

s

(

R

3

) for any

t, s

[2

,

2

) then

Z

R3

[ (

Hu

n

(

x, un, vn

)

Hu

(

x, u, v

))(

un

u

) ]

dx

+

Z

R3

[ (

Hv

n

(

x, un, vn

)

Hv

(

x, u, v

)(

vn

v

)) ]

dx

0

,

as

n

→ ∞

.

proof of theorem 1.3

By proposition 3.2 the functional

I

(

u, v

) satisfies

in Cerami condition. Now, we show that

I

(

u, v

) satisfies in condition

I

1

and

I

2

in theorem 1.2. From

H

2

,

lim

|(u,v)|→∞

H

(

x, u, v

)

|

(

u, v

)

|

4

= +

,

for any

M >

0 there exists

N >

0 such that

x

R

3

,

|

(

u, v

)

| ≥

N,

H

(

x, u, v

)

1

4

M

|

(

u, v

)

|

4

,

for any

x

R

3

and (

u, v

)

R

2

.

Therefore,

I

(

u, v

) =

1

2

k

(

u, v

)

k

2 XE

+

1

4

Z

R3

[

φuu

2

+

ψvv

2

]

dx

Z

R3

H

(

x, u, v

)

dx

1

2

k

(

u, v

)

k

2 XE

+ [

M

4

4

(

k

u

k

4

E

+

k

v

k

4E

) ]

M

4

k

(

u, v

)

k

4

XE

+ ˜

M

k

(

u, v

)

k

2 XE

,

where ˜

M

:=

sup

|(u,v)|<N

(

M4

|

(

u, v

)

|

4

H(x,u,v)

|(u,v)|2

)

.

Then

H

(

x, u, v

)

M

4

k

(

u, v

)

k

4

XE

M

˜

k

(

u, v

)

k

2 XE

.

Hence,

I

(

u, v

)

1

2

k

(

u, v

)

k

2 XE

+

M

4

4

(

k

u

k

4

E

+

k

v

k

4E

)

M

12M

4

[

k

u

k

4

E

+

k

v

k

4E

]

+

M

12

M

˜

(

k

u

k

2E

+

k

v

k

2E

)

,

where

M

12

is positive constant. Since,

M44

M124M

<

0

,

so for

M

large

enough, it follows that

a

k

:=

max

(u,v)∈Wk,k(u,v)kXE=ρk

I

(

u, v

)

0

,

for some positive constant large enough. Now, using the lemma 2.3 and

2.1 we show that

I

(

u, v

) satisfies in condition

I

2

.

I

(

u, v

) =

1

2

k

(

u, v

)

k

2 XE

+

1

4

Z

R3

[

φ

u

u

2

+

ψ

v

v

2

]

dx

Z

R3

(14)

1

2

k

(

u, v

)

k

2 XE

(

M

1

2

k

u

k

2 L2

+

M

1

p

k

u

k

p Lp

+

M

2

2

k

v

k

2 L2

+

M

2

p

k

v

k

p Lp

)

1

2

[

k

u

k

2

E

+

k

v

k

2E

]

M

1

C

emb

2

k

u

k

2 E

β

kp

M

1

C

emb

p

k

u

k

p E

M

2

C

emb

2

k

v

k

2 E

M

2

C

emb

α

pk

p

k

v

k

p E

= (

1

2

M

1

C

emb

2

)

k

u

k

2 E

+ (

1

2

M

2

C

emb

2

)

k

v

k

2 E

[

M

1

C

emb

β

kp

p

k

u

k

p E

+

M

2

C

emb

α

pk

p

k

v

k

p E

]

.

We choose

r

k

:= (

M1βCpemb k

+

M2Cemb

αpk

)

1

p−2

.

Hence,

b

k

:=

inf

(u,v)∈Zk,k(u,v)kXE=rk

I

(

u, v

)

inf

(u,v)∈Zk,k(u,v)kXE=rk

[ (

1

2

M

1

C

emb

2

)

k

u

k

2 E

M

1

Cembβ

kp

p

k

u

k

p E

]

+(

1

2

M

2

C

emb

2

)

k

v

k

2 E

M

2

Cembα

pk

p

k

v

k

p E

}

inf

(u,v)∈Zk,k(u,v)kXE=rk

M

13

(

k

u

k

2E

+

k

v

k

2E

)

M

14

(

k

u

k

pE

+

k

v

k

pE

)

,

where

M

13

:=

min

{

1−M12Cemb

,

1−M22Cemb

}

and

M

14

:=

min

{

1−M1βpkCemb

p

,

1−M2αpkCemb

p

}

are positive constants. Therefore,

I

(

u, v

)

inf

(u,v)∈Zk,k(u,v)kXE=rk

M

13

k

(

u, v

)

k

2XE

M

14

k

(

u, v

)

k

p

XE

M

15

r

2 k

(1

r

p−2 k

)

,

where

M

15

:=

min

{

M

13

, M

14

}

is positive constant. Now, if

k

→ ∞

,

then by lemma 3.1 we have

rk

+

.

Therefore, by theorem 1.2 the

system 1.1 has infinitely many solutions.

References

[1] V. Benci, D. Fortunato, An eigenvalue problem for the Sch¨o rdinger-Maxwell equations ,Topol.Methods nonlinear Anal.11(2)(1998), 283-293. [2] J. Zhang, W. Qin, F. Zhao, Existence and multiplicity of solu-tions for asymptotically linear nonpreiodic Hamiltonian elliptic system,

J.Math.Anal.Appl.399(2013), 433-441.

[3] V. Bence, P.H. Rabinowitz , Critical point theorems for indefinite func-tionals , Invent. Math.52 (1979), 241-273.

[4] D.G. DeFigueiredo, P.L. Felmer, On superquadiatic elliptic systems,

Trans.Amer.Math.Soc .343 (1994) 97-116.

[5] D.G. DeFigueiredo, J.M. Do´o, B. Ruf, An Orlicz-space approch to super-linear elliptic systems,J.Funct.Anal. 224(2004), 471-496.

[6] L. Li, Sh-J. Chen, Infinitely many large energy solutions of super-linear Schr¨odinger-maxwell equations, Electron.j.Differential Equations.

Vol.(2012) No. 224 (2012), 1-9.

[7] D. Bonheure, J.Di Cosmo, J.Van Schaftingen, Nonlinear Schrdinger equa-tion with unbounded or vanishing potentials: Soluequa-tions concentrating on lower dimensional spheres,J.Differential Equations.252(2012) , 941-968. [8] A. Ambrosetti, A. Malchiodi, Perturbation Methods and Semilinear

(15)

[9] Th.Bartsch, Sh. Peng, Semiclassical symmetric Schrdinger equations: ex-istence of solutions concentrating simultaneously on several spheres , Z. Angew. Math. Phys. 58(5)(2007), 778-804.

[10] D. Bonheure, J. Van Schaftingen, Bound state solutions for a class of nonlinear Schrdinger equations,Rev. Mat. Iberoam. 24 (2008), 297-351. [11] S.J.Chen, C.L. Tang, High energy solutions for the superlinear

Schr¨odinger-Maxwell ewuations, Nonlinear Anal. 71(10) (2009), 4927-4934.

[12] G. Cerami, An existence criterion for the points on unbounded manifolds, Istit.Lombardo Accad. Sci.Lett.Rend.A .112(2)(1979), 332-336.

[13] W.m. Zou, M. Schecheter, Critical point theory and its applications, Springer, New York, 2006.

[14] V. Benci, D. Fortunato, A.Masiello, L. Pisani, Solitons and the electro-magneticfield, Math.Z.232(1)(1999), 73-102.

References

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