Ground States for a Class of Nonlinear Schrodinger-Poisson Systems with Positive Potential ()
1. Introduction
In this paper, we consider the following nonlinear Schrodinger-Poisson systems
(1.1)
where
,
;
,
and
are positive potentials defined in
.
In recent years, such systems have been paid great attention by many authors concerning existence, non- existence, multiplicity and qualitative behavior. The systems are to describe the interaction of nonlinear Schrodinger field with an electromagnetic field. When
,
,
, the existence of non- trivial solution for the problem (1.1) was proved as
in [1] , and non-existence result for
or
was proved in [2] . When
,
,
, using critical point theory, Ruiz [3] obtained some multiplicity results for
, and existence results for
. Later, Ambrosetti and Ruiz [4] , and Ambrosetti [5] generalized some existence results of Ruiz [3] , and obtained the existence of infinitely solutions for the problem (1.1).
In particular, Sanchel and Soler [6] considered the following Schrodinger-Poisson-Slater systems
(1.2)
where
. The problem (1.2) was introduced as the model of the Hartree-Foch theory for a one-compo- nent plasma. The solution is obtained by using the minimization argument and
as a Lagrange multiplier. However, it is not known if the solution for the problem (1.2) is radial. Mugani [7] considered the following generalized Schrodinger-Poisson systems
(1.3)
where
,
and
, and proved the existence of radially symmetric solitary waves for the problem (1.3).
In this paper, without requiring any symmetry assumptions on
,
and
, we obtain the existence of positive radial ground state solution for the problem (1.1). In particular, the positive potential
implies that we are dealing with systems of particles having positive mass. It is interesting in physical applications.
The paper is organized as following. In Section 2, we collect some results and state our main result. In Section 3, we prove some lemmas and consider the problem (1.1) at infinity. Section 4 is devoted to our main theorem.
2. Preliminaries and Main Results
Let
,
denotes a Lebesgue space, the norm in
is
, ![]()
is the completion of
with respect to the norm
![]()
be the usual Sobolev space with the usual norm
.
Assume that the potential
satisfies
H1)
,
,
.
Let
be the Hilbert subspace of
such that
(2.1)
Then
,
with the corresponding embeddings being continuous (see [8] ). Furthermore, assume the potential
satisfies
H2)
,
,
.
It is easy to reduce the problem (1.1) to a single equation with a non-local term. Indeed, for every
, we have
(2.2)
Since
,
and (2.1), by the Lax-Milgram theorem, there exists a
unique
such that
(2.3)
It follows that
satisfies the Poisson equation
![]()
and there holds
![]()
Because
, we have
when
, and
,
is positive constant.
Substituting
in to the problem (1.1), we are lead to the equation with a non-local term
. (2.4)
In the following, we collect some properties of the functional
, which are useful to study our problem.
Lemma 2.1. [9] For any
, we have
1)
is continuous, and maps bounded sets into bounded sets;
2) if
weakly in
, then
weakly in
;
3)
for all ![]()
Now, we state our main theorem in this paper.
Theorem 2.2. Assume that
,
, the potential
satisfies condition H1), the potential
satisfies condition H3) and
, the potential
satisfies
H3)
,
, ![]()
and
,
on positive measure. Then there exists a positive radial ground state solution for the problem (1.1).
Remark 2.3. If
,
,
and
are positive potentials defined in
, and
,
be a solution for the problem (1.1). Then
, Indeed, we have
![]()
Since
, this implies
. By Lemma 2.1, we have
.
3. Some Lemmas and the Problem (1.1) at Infinity
Now, we consider the functional
given by
![]()
Since
satisfies condition H2), by (2.2), the Holder inequality and Sobolev inequality, we have
, (3.2)
where
and
. Since the potential
satisfies condition Q,
, we have
![]()
By Sobolev inequality, we obtain that
(3.3)
Combining (3.2) and (3.3), we obtain that the functional
is a well defined
functional, and if
is critical point of it, then the pair
is a weak solution of the problem (1.1).
Now, we define the Nehari manifold ([10] ) of the functional ![]()
,
where
![]()
Hence, we have
(3.4)
Lemma 3.1. 1) For any
,
, there exists a unique
such that
. Moreover, we have ![]()
2)
is bounded from below on
by a positive solution.
Proof. 1) Taking any
and
, we obtain that there exists a unique
such
that
. Indeed, we define the function
. We note that
if only if
. Since
is equivalent to
.
By
,
and
, we have
.
By
,
, the equation
has a unique
and the corresponding point
and
.
2) Let
, by (3.4) and
, we have
![]()
By the definition of Nehari manifold
of the functional
, we obtain that
is a critical point of
if and only if
is a critical point of
constrained on
(3.5)
Now, we set
![]()
By 2) of Lemma 3.1, we have ![]()
Since
,
,
, we consider the problem (1.1) at infinity
(3.6)
Similar to (2.2), we obtain that there exists a unique
such that
.
It follows that
satisfies the Poisson equation
(3.7)
Hence substituting
into the first equation of (3.6) we have to study the equivalent problem
(3.8)
The weak solution of the problem (3.8) is the critical point of the functional
![]()
where
is endowed with the norm
![]()
Define the Nehari manifold of the functional ![]()
,
where
![]()
and
![]()
The Nehari manifold
has properties similar to those of ![]()
Lemma 3.2. The problem (3.8) has a positive radial ground state solution
such that
![]()
For the proof of Lemma 3.2, we make use of Schwarz symmetric method. We begin by recalling some basic properties.
Let
such that
, then there is a unique nonnegative function
, called the Schwarz symmetric of
, such that it depends only on
, whose level sets
.
We consider the following Poisson equation
![]()
From Theorem 1 of [11] , we have
.
Hence, let
,
and
,
, we have
. (3.9)
The Proof of Lemma 3.2. Let
be such that
Let
such that
then we have
,
and
.
Hence, we obtain that
. (3.10)
Since
and
, (3.10) implies that
. Therefore, we can assume that
.
On the other hand, let
be the Schwartz symmetric function associated to
, then we have
(3.11)
Let
be such that
, and
, by (3.9) and (3.11), we have
![]()
This implies that
. Therefore, we have
, and we can suppose that
is radial
in
. Since
is compactly embedded into
for
, we obtain that
is achieved at some
which is positive and radial. Therefore, Lemma 3.2 is proved.
4. The Proof of Main Theorem
In this section, we prove Theorem 2.2. Firstly, we consider a compactness result and obtain the behavior of the (PS) sequence of the functional
.
Lemma 4.1. Let
be a (PS)d sequence of the functional
constrained on
, that is
(4.1)
Then there exists a solution
of the problem (2.4), a number
,
functions
of
and
sequences of points
,
such that
1)
,
, if
,
;
2)
;
3)
;
4)
are non-trivial weak solution of the problem (3.8).
Proof. The proof is similar to that of Lemma
4.1 in
[9] .
By Lemma 4.1, taking into account that
for all
and
, we obtain that ![]()
and
in
(strongly), i.e.
is relatively compact for all
. Hence we only need to prove that the energy of a solution of the problem (2.4) cannot overcome the energy of a ground state solution of the problem (3.8).
The proof of Theorem 2.2. By Lemma 4.1, we only prove that
. Indeed, let
such that
, and let
such that
. Since
,
and
, we have
(4.2)
Since
and
, we have
![]()
Therefore, we have
![]()
By
, we have
. If
, we have
and
. Hence, by
, we have
(4.3)
and by
, we have
. (4.4)
Combining (4.3) and (4.4), we have
![]()
Since
,
,
, and
on a positive measure, we have
![]()
which is not identically zero, and is contradiction. Hence, we have
. By (4.2), we have
![]()
Then there exists a positive radial ground state solution for the problem (1.1).
Acknowledgements
This research is supported by Shanghai Natural Science Foundation Project (No. 15ZR1429500), Shanghai Leading Academic Discipline Project (No. XTKX2012) and National Project Cultivate Foundation of USST (No. 13XGM05).
NOTES
*Corresponding author.