Existence and Stability of Standing Waves for the Nonlinear Schrödinger Equation with Combined Nonlinearities and a Partial Harmonic Potential ()
1. Introduction
In this paper, we study the existence and stability of standing waves for the nonlinear Schrödinger equation with combined power-type nonlinearities and a partial harmonic potential
(1.1)
where
is a complex-valued function of
,
,
,
. The external potential W describes the
electromagnetic trap for the condensate and is usually chosen to be an istropic quadratic confinement, i.e.,
where
represent the corresponding trap frequency in each spatial direction. The Gross-Pitaevskii equation with N particle bodies are strictly derived by Gross and Pitaevskii, see [1] [2] . Due to the inclusion of a quadratic potential, the natural energy space for studying Equation (1.1) is given by the following expression
with the norm
Different forms of the potential W correspond to different physical meanings. In the case
, Equation (1.1) arises in various areas of physics and mathematics. The typical class of nonlinear dispersion equation had been proposed by Schrödinger in [3] . When
,
, it is a nonlinear Schrödinger equation with harmonic limiting potential and W indicates that the external potential is uniformly distributed in all directions of space in a harmonic form. This type of equations has been studied in [4] [5] [6] . In fact, when
,
, some coefficients of
will vanish, the harmonic potential becomes partial harmonic potential. Then Equation (1.1) does not keep invariant by translation. The orbital stability of standing waves for the inhomogeneous Gross-Pitaevskii equation has been studied in [7] .
In [4] , Carles and Il’yasov considered the nonlinear Schrödinger equation with a harmonic potential in the presence of two combined energy-subcritical power nonlinearities. They address the equations of the existence and the orbital stability of the set of standing waves by the method of fundamental frequency solutions. This method makes it possible to describe accurately the set of fundamental frequency standing waves and ground states, and to prove its orbital stability. On this basis, this paper converts the harmonic potential into a partial harmonic potential. At this time, the compactness disappears, which makes it more difficult to discuss the stability of the standing waves for the equation. Therefore, we must seek new ways to address the issues raised in this article.
Equation (1.1) enjoys a class of special solutions, which are called standing waves, namely solutions of the form
, where
, and the function
solves the following elliptic equation
(1.2)
A possible choice is then to fix
, and to search for solutions to (1.2) as critical points of action functional
(1.3)
where the energy
is defined as
(1.4)
For the Equation (1.1), an important issue is to consider the stability of standing waves, which is defined as follows:
Definition 1.1. A set
is orbitally stable if for any given
, there exists
such that for any initial data
satisfying
the corresponding solution
of (1.1) satisfies
Given this definition, for the sake of stability, we require that the solution of (1.1) exists globally, at least for initial data
close enough to
. In L2-supercritical case, according to the local well-posedness theory of NLS, small initial data NLS solution exists globally, while for some large initial data, the solution may blow up in finite time. Therefore, it is especially important to pay attention to whether there is a stable standing waves in this case.
In order to study the orbital stability of standing waves, we apply the idea by Cazenave and Lions in [8] , consider the following constrained minimization problem
where
However, since the nonlinearities is the L2-supercritical, the energy functional is unbounded from below on
. Indeed, when
, taking
such that
, then we have
(1.5)
as
. Therefore, we cannot study the existence and stability of standing waves of Equation (1.1) by considering the global minimization problem. Due to this type of problems has been considered in [9] [10] [11] by studying the corresponding local minimization problems, we consider the following local minimization problem: for any given
, defining
(1.6)
where
and
is given by
(1.7)
It can be proved that for any given
with some
, there exists a
, such that
. Thus we can obtain the existence of a minimizer of
. Denote the set of all minimizers of
by
To prove the existence and stability, the key is to show that any minimizing sequence is relatively compact. For Equation (1.1), when
, the
embedding
↪
with
is compact, the minimization problem (1.6) can be easily solved. However, when some of coefficients of
vanish, the embedding
↪
with
is not compact. In this
case, the general method is to apply concentration compactness principle to overcome this difficulty. Then we can obtain the compactness of all minimizing sequences of (1.6) and prove the existence and stability of standing waves for (1.1). Without loss of generality, we assume
(1.8)
where
.
According to (1.8), our main results are as follows:
Theorem 1.2. Let
,
,
, all being
fixed, then there exists
, such that for every given
, there exists
with
, we have for any
that
1)
;
2) The set
is orbitally stable.
This paper is organized as follows: in Section 2, we given some preliminary results, which will be used later. In Section 3, we prove Theorem 1.2.
2. Preliminaries
In this section, we recall some preliminary results that will be used later. Firstly, let us recall the local well-posedness theory for the Cauchy problem (1.1) established in [12] .
Lemma 2.1. Let
,
,
, and
.
Then, there exists
, such that (1.1) admits a unique solution
. Let
be the maximal time interval on which the solution
is well-defined, if
, then
as
. Moreover, for all
, the solution
satisfies the following conservations of mass and energy
and
where
is defined by (1.4).
Lemma 2.2. [13] Define
and
Then
.
Lemma 2.3. [14] Let
and
, then the following sharp Gagliardo-Nirenberg inequality
holds for any
. The sharp constant
is
where Q is defined in Theorem 1.4 by [8] .
Lemma 2.4. [9] Let
, suppose that
almost everywhere and
is a bounded sequence in
, then
3. Proof of Theorem 1.2
In this section, we first establish a local minima structure for
on
.
Lemma 3.1. Let
,
,
, all
being fixed, then there exists
, such that for every given
, there exists
with
, we have
(3.1)
(3.2)
Proof. Let
be such that
,
. Then for all
, letting
, we have
and
namely
. Thus (3.1) is verified.
To verify (3.2), using the Gagliardo-Nirenberg inequality, we have
Denote
, we define the following functions:
Notice that for any
, there exists
, such that for all
with
, we have
(3.3)
This implies that
which completes the proof.
Lemma 3.2. Let
,
and
. Let
and
be as Lemma 3.3. Let
and
be a minimizing sequence of (1.6). Then there exists
such that
(3.4)
Proof. If there exists a subsequence, still denoted by
such that
as
, then by the interpolation,
as
, for
all
,
. We consequently obtain that
(3.5)
where
is defined by Lemma 2.2.
On the other hand, since the space
is compactly embedding in
, it is standard to show that
is achieved by some
with
. Let
satisfy
, and set
where
Then
for all
. It follows that
(3.6)
for
small enough. Notice that
for
sufficiently small, we consequently obtain that
This is a contradiction with (3.8). This completes the proof.
Lemma 3.3. Let
,
and
. Let
and
be as Lemma 3.1. Then for any
, we have
(3.7)
Proof. We first prove the following strict monotonicity:
(3.8)
Indeed, let
satisfy
. Then by Lemma 3.2, there exists a
such that
Let
, we have
. Then we get
(3.9)
which implies (3.11). Then for all
, we get
(3.10)
Proof of Theorem 1.2. Let
be a minimizing sequence of
, namely
Applying Lemma 3.2, there exist
and
, such that
where
. We first prove
. If not, denote
and
. Appling Brezis-Lieb Lemma, we have
and
Then we see that
,
. Choosing a subsequence of
(still denote by
) such that
, we deduce that
(3.11)
which is a contradiction with Lemma 3.3. Therefore
and
strongly in
. Then by using the interpolation, we get that
strongly in
for all
. We then deduce from the weak convergence in X that
. On the other hand, due to
, we have
. We consequently obtain
and then
. Thus, we deduce that
strongly in X, namely
strongly in X and
.
Next we prove that
is orbitally stable by contradiction. We assume that there exist
and a sequence of initial data
such that
(3.12)
and there exist a sequence
such that the maximal solution
with
satisfies that
(3.13)
Without restriction, we can assume
such that
is a minimizing sequence of (1.6). According to Lemma 3.1, when n is sufficiently large,
we have
, which together with
and
, implies that
is a minimizing sequence for (1.6). Then we have
by Lemma 3.1. This
shows that
is globally large for sufficiently large n. Due to the compactness of all minimizing sequence of (1.6), a contradiction to (3.16) is obtained. Theorem 1.2 has been proven.
4. Conclusion
In recent years, the nonlinear Schrödinger equation have been studied by many experts. This paper mainly adds a nonlinear term and a partial harmonic potential on this basis. In particular, the addition of nonlinear terms poses significant computational challenges, as the equations lose their compactness and translation invariance due to the presence of partial harmonic potentials. To solve the difficulty, we were inspired by the [8] , the compactness of the minimization sequence is obtained by establishing the minimalization problem and using the concentration compactness principle, thus proving the stability of the standing waves for the equation in L2-supercritical case.