Normalized Solutions of Mass-Subcritical Schrödinger-Maxwell Equations ()
1. Introduction
In this paper, we study the existence of normalized ground state solution of the following Schrödinger-Maxwell equations
(1.1)
where
and
, the parameter
appears as a Lagrange multiplier. The unknowns of the equations are the field u associated to the particle and the electric potential
, and satisfying the normalization condition
(1.2)
we prescribe
. Hence, we have
(1.3)
where u belongs to the Hilbert space
and
The space
is endowed with the norm
Let
with respect to the norm
For any
,
is endowed with the norm
Obviously, the embedding
↪
is compact (see [1] ).
By the variational nature, the weak solutions of (1.1) are critical points of the functional
defined by
where
is a rather general nonlinearity. Then, it is clear that the function J is
on
and has the strong indefiniteness. We can know that the weak solutions of (1.1)
are critical points of the functional J. By standard arguments, the function J is
on
.
In recent years, normalized solutions of Schrödinger equations have been widely studied. When searching for the existence of normalized solutions of Schrödinger equations in
, appears a new mass-critical exponent
Now, let us review the involved works. In the mass-subcritical case, Zuo Yang and Shijie Qi [2] proved that for all
, the following Schrödinger equations with potentials and non-autonomous nonlinearities
have a normalized solutions. Nicola Soave [3] in the mass-subcritical proved the nonlinear Schrödinger equation with combined power nonlinearities mass- critical and mass-supercritical cases studied of:
have several stability/instability and existence/non-existence results of normalized ground state solutions. For
is a superlinear, subcritical, Thomas Bartsch [4] studied the existence of infinitely many normalized solutions for the problem
By establishing the compactness of the minimizing sequences, Tianxiang Gou and Louis Jeanjean [5] in the mass-subcritical studied the existence of multiple positive solutions to the nonlinear Schrödinger systems:
In the mass-subcritical case, Masataka Shibata [6] studied for the nonlinear Schrödinger equations with the minimizing problem:
where
is a general nonlinear term. They proved
is attained. That is to say, the Schrödinger equations have normalized solutions.
Moreover, for the
case, Norihisa
Ikoma and Yasuhito Miyamoto [7] showed the existence of the minimizer of the minimization problem
, where
as
. They also obtained the conclusions that the normalized solutions of Schrödinger equations exist. In the mass-subcritical condition, Zhen Chen and Wenming Zou [8] basing on the refined energy estimates proved the existence of normalized solutions to the Schrödinger equations.
Other related normalized solutions problems of Schrödinger can be seen in [9] [10] [11] [12] [13] . Thus, the main purpose of this paper is to study the solution of Schrödinger-Maxwell equations satisfying normalization condition by using above results. In particular, the situation we consider will involve the presence of potential
. In addition, the nonlinear term
is mass-sub- critical and satisfies the following appropriate assumptions. In this case, the functional I is bounded from below and coercive on
, which will be proved in Lemma 2.5.
We assume the following conditions throughout the paper:
(f1)
is continuous.
(f2)
and
with
.
Moreover, c and
are positive constants which may change from line to line.
Our main result is the following theorem:
Theorem 1.1 Suppose (f1) and (f2) hold. Then, for any
, problem (1.3) has a normalized ground state solution.
2. Proof of Main Results
Since the functional J exhibits a strong indefiniteness. To avoid the difficulty we use the reduction method. Thus, we shall introduce the method.
For any
, us consider the linear operator
defined as
(2.1)
Then, there exists a positive constant
such that
because the following embeddings are continuous:
↪
,
and
↪
We set
Obviously,
is linear in
and v respectively.
Moreover, there exists a positive constant
and
such that for any
,
(2.2)
(2.3)
Combining (2.2) and (2.3) we know that
is bounded and coercive. Hence, by the Lax-Milgram theorem we have that for every
, for any
, there exists a unique
such that
Then, for any
, we obtain
(2.4)
and using integration by parts, we have
Therefore,
(2.5)
in a weak sense, and
has the following integral expression:
(2.6)
The functions
possess the following properties:
Lemma 2.1 For any
, we have:
1)
, where
is independent of u. As a consequence there exists
such that
2)
.
Proof. 1) For any
, using (2.5) we have
where
is a positive constant. Hence, we obtain that
therefore there exists a positive constant
such that
(2.7)
because we know for any
,
↪
.
2) Obviously, by the expression (2.6) the conclusion holds. □
Now let us consider the functional
,
Then I is
.
By the definition of J, we have
Multiplying both members of (2.5) by
and integrating by parts, we obtain
Therefore, the functional I may be written as
(2.8)
The following lemma is Proposition 2.3 in [5] .
Lemma 2.2 The following statements are equivalent:
1)
is a critical point of J.
2) u is a critical point of I and
.
Hence u is a solution to (1.3) if and only if u is the critical point of the functional (2.8). The critical point can be obtained as the minimizer under the constraint of
-sphere
(2.9)
We shall study the constraint problem as follows:
(2.10)
The solution of (13)
is called a normalized ground state solution satisfying problem (3) if it has minimal energy among all solutions:
In this paper, we will be especially interested in the existence of normalized ground state solutions.
Lemma 2.3 We define
,
, which is also the solution of the Equation (2.5) in
. Let
be a minimizing sequence of I with satisfying
in
. Then,
in
and we obtain
as
(2.11)
Proof. By (2.1), the following expressions hold
Since
and the embedding
↪
is compact for any
, clearly we have
(2.12)
then, by interpolation we have
Using again (2.12), we get
Moreover,
be a minimizing sequence and
in
, we obtain
(2.13)
Therefore, we get
which implies that
converges strongly to
.
Hence, we obtain
(2.14)
By (2.13) and (2.14), we know that conclusion (2.11) holds. □
Lemma 2.4 (Gagliardo-Nirenberg inequality). For all
, we have
where
is a positive constant depending on N and
.
Lemma 2.5 Suppose (f1) and (f2) hold, than for any
, the functional I is bounded from below and coercive on S(a).
Proof. Assumptions (f1) and (f2) imply that for any
, there exist
such that
Hence, according to Lemma 2.4 with
, we obtain that
Choose
such that
, than
Therefore, I is bounded from below and coercive on
. □
The following lemma is Lemma 2.2 in [6] .
Lemma 2.6 Suppose (f1) and (f2) hold and
is a bounded sequence in
. If
holds, then it is true that
Next, we collect a variant of Lemma 2.2 in [14] . The proof is similar, so we omit it.
Lemma 2.7 Suppose (f1) and (f2) hold and
is a bounded sequence in
, then we have
in
, thus
Proof of Theorem 1.1. Let
be a minimizing sequence of I with concerning
. Then, by (9) we obtain
According to Lemma 2.5, the sequence
is bounded in
. Letting
be in
. Moreover, we know that the embedding
↪
is compact. Hence, we conclude
(2.15)
(2.16)
We also have
Since (19) holds, we have
. Then, by Lesmma 2.6 we obtain
Moreover, by Lemma 2.7 we have
which implies
(2.17)
Hence, combining weak lower semicontinuity of the norm
, Lemma 2.3 and (2.17), we have
which implies
. Then,
satisfies
and
. Therefore, problem (1.3) has a normalized ground state solution. □