Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 76, pp. 1–20.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2023.76

STABILITY OF GROUND STATES OF NONLINEAR

SCHRÖDINGER SYSTEMS

LILIANA CELY

Abstract. In this article, we study existence and stability of ground states
for a system of two coupled nonlinear Schrödinger equations with logarith-

mic nonlinearity. Moreover, global well-posedness is verified for the Cauchy

problem in H1(R) ×H1(R) and in an appropriate Orlicz space.

1. Introduction

We consider the coupled system of logarithmic nonlinear Schrödinger equations

i∂tu+ ∂2
xu+ u log |u|2 + u|u|p−1|v|p+1 = 0

i∂tv + ∂2
xv + v log |v|2 + v|v|p−1|u|p+1 = 0

u(x, 0) = u0(x) v(x, 0) = v0(x),

(1.1)

where u = u(x, t) and v = v(x, t) are complex-valued functions of (x, t) ∈ R × R,
and 1 ≤ p < 2.

The motivation for studying coupled NLS equations mainly comes from their
applications in various physical fields; for example, in quantum optics, quantum
mechanics, nuclear physics, fluid dynamics, plasma physics and Bose-Einstein con-
densation (for the principal references see [26, 25]). Such systems (1.1) appear in
the study of interactions between short and long dispersive waves (see [1, 14, 14]).
Moreover, numerical studies describing the dynamics of gaussons collisions of (1.1)
were reported in [23]. For local well-posedness in the energy space, existence and
stability of standing waves for coupled nonlinear Schrödinger system with power-
type nonlinearities, the reader is referred to [2, 11, 12, 15, 20, 21, 5, 6, 7, 22, 3, 4]
and references therein.

System (1.1) has the conserved quantities

E(u, v)

=
1

2

∫
R

{
|∂xu|2 + |∂xv|2 − |u|2 log |u|2

− |v|2 log |v|2 − 2

p+ 1
|u|p+1|v|p+1

}
dx (1.2)

2020 Mathematics Subject Classification. 35Q55, 35Q40.
Key words and phrases. Nonlinear Schrödinger system; global well-posedness; stability;

ground states.
©2023. This work is licensed under a CC BY 4.0 license.

Submitted March 16, 2023. Published November 1, 2023.

1



2 L. CELY EJDE-2023/76

and

Q(u, v) =
1

2

∫
R

(|u|2 + |v|2) dx. (1.3)

That is, when applied to sufficiently regular solutions (u(x, t), v(x, t)) of 1.1, the
energy functional E and the mass Q are independent of t.

Notice that the function z 7→ |z|2Log|z|2 has a singularity at the origin, and
hence the energy functional E(u, v) fails to be differentiable on H1(R) × H1(R).
Thus, to overcome the singularity of the logarithm at the origin, we need to define
an energy space where the functional E is well-defined. Indeed, we define the energy
space B := W (R)×W (R),

W (R) = {u ∈ H1(R) : |u|2 log |u|2 ∈ L1(R)},
endowed with a Luxemburg type norm. In Section 2 we show that the energy
functional E is of class C1 on B. Now we state our first result concerning the global
well-posedness of the Cauchy problem (1.1). The proof is contained in Section 2.

Theorem 1.1. Assume that 1 ≤ p. For every (u0, v0) ∈ B the Cauchy problem
(1.1) is locally well posed in B, i.e. there exist T > 0 and a unique solution (u, v) ∈
C([0, T ],B) such that (u(x, 0), v(x, 0)) = (u0, v0). For each T0 ∈ (0, T ) the mapping
(u0, v0) ∈ B 7→ (u, v) ∈ C([0, T0],B) is continuous. In addition, the mass associated
with (1.1), as well as energy are conserved in time, namely, for all [0, T ]

‖u(x, t)‖2L2 + ‖v(x, t)‖2L2 = ‖u0‖2L2 + ‖v0‖2L2 , E(u(x, t), v(x, t)) = E(u0, v0).

From a mathematical and physical point of view, an important type of solutions
for the system (1.1) are the so-called standing waves. In this article we are interested
in the existence and stability of standing waves of (1.1); namely solutions to (1.1)
of the form

(f(x, t), g(x, t)) = (eiθtφ(x), eiωtϕ(x)), θ, ω ∈ R and (φ, ϕ) ∈ B,
where (φ, ϕ) has to satisfy the system of ordinary differential equations

−∂2
xφ+ θφ = φ log |φ|2 + φ|φ|p−1|ϕ|p+1

−∂2
xϕ+ ωϕ = ϕ log |ϕ|2 + ϕ|ϕ|p−1|φ|p+1.

(1.4)

The most common approach to construct orbitally stable standing waves to (1.1)
is to consider the variational problem

J(η, ζ) := inf
{
E(u, v) : (u, v) ∈ B,

∫
R
|u|2 = η and

∫
R
|v|2 = ζ

}
. (1.5)

A minimizer of problem (1.5) is called a ground state solution of (1.4). The corre-
sponding set of (non-trivial) minimizers for J(η, ζ) is defined by

G(η, ζ) = inf{(φ, ϕ) ∈ B : E(φ, ϕ) = J(η, ζ), ‖φ‖2L2 = η and ‖ϕ‖2L2 = ζ}.
From the logarithmic Sobolev inequality, it is not difficult to show that problem
(1.5) is well-defined; that is J(η, ζ) > −∞ (see Lemma 3.1 below). Moreover, by
using the concentrated compactness principle [18], the existence of ground states
will be obtained as a consequence of the stronger statement that any minimizing
sequence for the problem J(η, ζ) is, up to translation, precompact in B. More
precisely, we have the following result.

Theorem 1.2. Let η > 0, ζ > 0 and 1 ≤ p < 2. Then the following assertions
hold.



EJDE-2023/76 NONLINEAR SCHRÖDINGER SYSTEMS 3

(i) If {(fn, gn)} is a minimizing sequence of J(η, ζ), then there exists a se-
quence {yn} of real numbers such that {(fn(· + yn), gn(· + yn))} contains
a convergent subsequence in B. Which means that {(fn, gn)} is relativity
compact in B up to translations. Hence, the set G(η, ζ) is non-empty, since
there exists a minimizer for problem (1.5).

(ii) If (f, g) ∈ G(η, ζ), then there exist θ and ω ∈ R such that f(x) = eiθφ(x)
and g(x) = eiωϕ(x), where φ, ϕ ∈ C2(R) and φ(x), ϕ(x) > 0 for all x ∈ R.

(iii) The set of minimizers G(η, ζ) forms a true two-parameter family; that is,
the two sets G(η1, ζ1) and G(η2, ζ2) are disjoint if (η1, ζ1) 6= (η2, ζ2).

To prove Theorem 1.2, we will use variational methods and the concentration
compactness method of P.L. Lions [18]. Similar techniques have been used previ-
ously by Albert and Bhattarai [2] (see also[21, 7]) to prove the existence and orbital
stability of standing wave solutions to NLS-KdV systems.

It is standard that the minimizers of the variational problem (1.5) are solutions
to the stationary problem (1.4). The following is our orbital stability result, which
is a direct consequence of the precompactness of the minimizing sequences of (1.5)
(see Theorem 1.2). This result shows that if the initial data of a solution of (1.1)
is near G(η, ζ), then the solution will remain near G(η, ζ) for every time t ≥ 0.

Corollary 1.3. Let η, ζ > 0. Then the set G(η, ζ) of minimizers for J(η, ζ) is B-
stable in the following sense. Given ε > 0 there exist δ > 0 such that, if (f0, g0) ∈ B
satisfies

inf
(ϕ,φ)∈G(η,ζ)

‖(f0, g0)− (ϕ, φ)‖B < δ

then the solution (f(x, t), g(x, t)) of the Cauchy problem (1.1) with initial data
(f(x, 0), g(x, 0)) = (f0(x), g0(x)) satisfies

inf
(ϕ,φ)∈G(η,ζ)

‖(f(·, t), g(·, t))− (ϕ, φ)‖B < ε

for all t ∈ [0,+∞).

To the best of our knowledge, this is the first work concerning the existence and
stability of ground states for the system (1.1) in the L2-subcritical case (1 ≤ p < 2).
We mention here that the well-posedness of the Cauchy problem (1.1), existence
and stability/ instability of standing waves is open problem in the L2-critical case
(p = 2) and L2-supercritical case (p > 2).

This article is organized as follows. In Section 2 we address the well-posedness
of Cauchy problem for system (1.1) and we give the proof of Theorem 1.1. Section
3 is devoted to study existence and stability of standing waves (Theorem 1.2 and
Corollary 1.3). Throughout this paper, the letter C will denote positive constants.

2. Existence results

This section is devoted to establish the local well-posedness of system (1.1). First
we need to introduce some notation. For every x ∈ [0,+∞), we define the functions
Φ(x), Ψ(x) ∈ C1([0,+∞)) ∩ C2((0,+∞)) as follows,

Φ(x) =

{
−x2 log(x2), if 0 ≤ x ≤ e−3;

3x2 + 4e−3x − e−6, if x ≥ e−3;

Ψ(x) = F (x) + Φ(x),

(2.1)



4 L. CELY EJDE-2023/76

where F (x) = x2Logx2. Notice that Φ is a nonnegative convex and increasing
function on [0,+∞). Next we define the Orlicz space LΦ(R) associated to Φ as the
completion of the C∞0 (R)-functions under the Luxemburg norm

‖f‖LΦ(R) = inf
{
k > 0 :

∫
R

Φ(k−1|f(x)|)dx ≤ 1
}
.

It is well-known that (see [8, Section 2]),

LΦ(R) = {f ∈ L1
loc(R) : Φ(|u|) ∈ L1(R)}.

From [8, Lemma 2.1] we see that (LΦ(R), ‖ · ‖LΦ(R)) is a reflexive Banach space. In
addition, we can prove the following properties.

Lemma 2.1. Let {fm} be a sequence in LΦ(R+), we have the following.

(i) If ‖fm−f‖LΦ(R) → 0 as n→ +∞, then ‖Φ(|fn|)−Φ(|f |)‖L1(R) as n→ +∞.

(ii) Let f ∈ LΦ(R). If fn(x)→ f(x) a.e. x ∈ R and if

lim
n→∞

∫
R

Φ(|fn(x)|)dx =

∫
R

Φ(|f(x)|)dx,

then ‖fm − f‖LΦ(R) → 0 as n→ +∞.

(iii) For each f ∈ LΦ(R), we have

min{‖f‖LΦ(R), ‖f‖2LΦ(R)} ≤
∫
R

Φ(|f(x)|)dx ≤ max{‖f‖LΦ(R), ‖f‖2LΦ(R)}. (2.2)

Also, notice that there exists a constant C > 0 such that (see [8, Eq. (2.6), p.
1131])∫

R
|Ψ(|f |)−Ψ(|g|)|dx ≤ C

(
1 + ‖f‖2H1(R) + ‖g‖2H1(R)

)
‖f − g‖L2(R), (2.3)

for all f, g ∈ H1(R). Finally, we define the reflexive Banach space

W (R) = H1(R) ∩ LΦ(R)

equipped with the usual norm ‖f‖W (R) = ‖f‖H1(R) + ‖f‖LΦ(R), for any f ∈W (R).
Combining (2.2) and (2.3), it is not hard to show that (see [8, Proposition 2.2])

W (R) = {u ∈ H1(R) : |u|2 log |u|2 ∈ L1(R)}.
It is important to note that the energy functional (1.2) is Fréchet differentiable on
B = W (R)×W (R). Moreover, the Fréchet derivative is

E′(f, g) =

[
−∂2

xf − f log |f |2 − f |f |p−1|g|p+1

−∂2
xg − g log |g|2 − g|g|p−1|f |p+1

]
(2.4)

for (f, g) ∈ B. The proof of (2.4) is similar to that of [8, Proposition 2.7] and we
omit here.

Now we sketch the basic points of the standard theory of Log NLS (see [9,
Chapter 9] and [8]) ensuring the local well-posedness of the Cauchy Problem for
(1.1) in the energy space B. First, we regularize the logarithmic nonlinearity near
the origin. Let z ∈ C and n ∈ N. Bearing in mind (2.1), we consider the functions

ζn(z) =

{
a(z), if |z| ≥ 1

n ;

n z a( 1
n ), if |z| ≤ 1

n ;
and βn(z) =

{
b(z), if |z| ≤ n;
z
nb(n), if |z| ≥ n,

where
a(z) =

z

|z|2
Φ(|z|), b(z) =

z

|z|2
Ψ(|z|) for z ∈ C, z 6= 0.



EJDE-2023/76 NONLINEAR SCHRÖDINGER SYSTEMS 5

We introduce the family of regularized nonlinearities in the form gn(z) = βn(z) −
ζn(z) for every z ∈ C. Notice that we have the point-wise limit gn(z) → z log |z|2
as n goes to +∞. Let (u0, v0) ∈ B (in particular u0, v0 ∈ H1(R)). For every n ∈ N,
we consider the initial value problem

i∂tu
n + ∂2

xu
n + gn(un) + un|un|p−1|vn|p+1 = 0

i∂tv
n + ∂2

xv
n + gn(vn) + vn|vn|p−1|un|p+1 = 0

un(x, 0) = u0(x) vn(x, 0) = v0(x).

(2.5)

Lemma 2.2. Assume that 1 ≤ p < 2. Then the initial value problem (2.5) is
well posed in H1(R) × H1(R), i.e., there exists a unique global solution (u, v) ∈
C(R, H1(R) × H1(R)) such that (u(0), v(0)) = (u0, v0). In addition, conservation
of energy and of L2-norm holds: for all t ∈ R,

‖un(t)‖2L2 + ‖vn(t)‖2L2 = ‖u0‖2L2 + ‖v0‖2L2 , En(un(t), vn(t)) = En(u0, v0)

where

En(u, v) =
1

2

∫
R

{
|∂xu|2 + |∂xv|2 −Gn(u)−Gn(v)− 2

p+ 1
|u|p+1|v|p+1

}
dx,

and

Gn(z) =

∫ |z|
0

gn(s) ds.

Proof. First, since gn is globally Lipschitz continuous C→ C, the proof of the local
wellposedness in H1(R)×H1(R) for 1 ≤ p < 2 is a contraction argument based on
Strichartz estimates; see Remark [9, 3.3.12] and [13, Theorem 1] for more details.

We need only show that the maximal solution of (2.5) is global. The proof
relies on the following Gagliardo-Nirenberg inequality [13, Section 3]: there exists
a constant C > 0 such that

‖u‖2p+2
L2p+2 + ‖v‖2p+2

L2p+2 + ‖uv‖p+1
Lp+1 ≤ C(‖∇u‖2L2 + ‖∇v‖2L2)p/2(‖u‖2L2 + ‖v‖2L2)p+1− p2

(2.6)
Indeed, by the local theory, we just need to control the H1(R) ×H1(R)-norm for
every t ∈ R. It is clear that

∫
RGn(u) dx ≤ C‖u‖2L2 . From (2.6), in view of the

conservation of charge and energy, we see

En(un, vn) ≥ 1

2
(‖∇un‖2L2 + ‖∇vn‖2L2)− C(‖un‖2L2 + ‖vn‖2L2)

− C(‖∇un‖2L2 + ‖∇vn‖2L2)p/2(‖un‖2L2 + ‖vn‖2L2)p+1− p2 .

Thus, we have

(‖∇un‖2L2 + ‖∇vn‖2L2)
[1
2
− C(‖∇un‖2L2 + ‖∇vn‖2L2)

p
2−1(‖un‖2L2 + ‖vn‖2L2)p+1− p2

]
≤ En(un, vn) + C(‖un‖2L2 + ‖vn‖2L2)

Since p < 2, by mass and energy conservation, we see easily that theH1(R)×H1(R)-
norm of the solution (un(t), vn(t)) is bounded. The continuity argument implies
that all solutions of (2.5) are global, which completes the proof. �

To show the uniqueness of the Cauchy problem (1.1), we will use Lemma 2.3
below.



6 L. CELY EJDE-2023/76

Lemma 2.3. (i) Let u, v, z, w ∈ C be such that |u|, |v|, |z|, |w| ≤ K, where K is
a positive constant. Then there exists C > 0 such that

|u|u|p−1|v|p+1 − z|z|p−1|w|p+1| ≤ C(|z − u|+ |w − v|).

(ii) For every z1, z2 ∈ C we have

| Im((z1 log |z1|2 − z2 log |z2|2)(z1 − z2))| ≤ 4|z1 − z2|2.

Proof. Statement (ii) follows immediately from [9, Lemma 9.3.5]. Next we prove
(i). A simple calculation shows that

|u||z|(u|u|p−1|v|p+1 − z|z|p−1|w|p+1)

= u|z|(|u|p|v|p+1 − |z|p|w|p+1) + [u(|z| − |u|) + |u|(u− z)]|z|p|w|p+1.
(2.7)

On the other hand, since p ≥ 1 and |u|, |v|, |z|, |w| ≤ K, it is not hard to show
that there exists a constant C > 0 such that∣∣|u|p|v|p+1 − |z|p|w|p+1big| ≤ C(|u− z|+ |v − w|). (2.8)

Therefore, combining (2.7) and (2.8) we infer that

|u|u|p−1|v|p+1 − z|z|p−1|w|p+1| ≤ ||u|p|v|p+1 − |z|p|w|p+1|+ 2|u− z||z|p−1|w|p+1

≤ C(|u− z|+ |v − w|),

and this completes the proof. �

Proof of Theorem 1.1. Our approach is inspired by Cazenave’s approach for the
single logarithmic Schrödinger equation in [9, Theorems 9.3.4, 3.3.5, 3.3.9].

Step 1. First, we regularize the logarithmic nonlinearity near the origin. Let z ∈ C
and n ∈ N. Bearing in mind (2.1), we consider the functions

ζn(z) =

{
a(z), if |z| ≥ 1

n ;

nza( 1
n ), if |z| ≤ 1

n ;
and βn(z) =

{
b(z), if |z| ≤ n;
z
nb(n), if |z| ≥ n,

where

a(z) =
z

|z|2
Φ(|z|), b(z) =

z

|z|2
Ψ(|z|) for z ∈ C, z 6= 0.

We introduce the family of regularized nonlinearities in the form gn(z) = βn(z) −
ζn(z) for every z ∈ C. Notice that we have the point-wise limit gn(z) → z log |z|2
as n→ +∞.

Further, we consider Jn = (I − 1
n∆)−1. Observe that Jn satisfies the following

properties

‖Jn‖L(H−1,H1) ≤ n, ‖Jn‖L(X,X) ≤ 1, (2.9)

where X coincides with one of the spaces H1(R), H−1(R) and L2(R). Moreover,

Jnu −→
n→∞

u in X for all u ∈ X;

if sup
n
‖un‖X <∞, then Jnu

n − un ⇀ 0 in X as n→∞. (2.10)

We define

g1,n(u, v) = Jng1(Jnu, Jnv), g2,n(u, v) = Jng2(Jnu, Jnv),

where

g1(u, v) = u|u|p−1|v|p+1, g2(u, v) = v|v|p−1|u|p+1.



EJDE-2023/76 NONLINEAR SCHRÖDINGER SYSTEMS 7

Let (u0, v0) ∈ B (in particular u0, v0 ∈ H1(R)). For every n ∈ N, we consider
the initial value problem

i∂tu
n + ∂2

xu
n + gn(un) + g1,n(un, vn) = 0

i∂tv
n + ∂2

xv
n + gn(vn) + g2,n(un, vn) = 0

un(x, 0) = u0(x), vn(x, 0) = v0(x).

(2.11)

Since gn, g1,n, g2,n are locally L2-Lipschitz continuous, by [9, Theorem 3.3.1], we
infer that there exists a unique global solution (un, vn) ∈ C(R, H1(R)×H1(R)) of
(2.11), for every n ∈ N, such that

‖un(t)‖2L2 + ‖vn(t)‖2L2 = ‖u0‖2L2 + ‖v0‖2L2 , En(un(t), vn(t)) = En(u0, v0) (2.12)

where

En(u, v) =
1

2

∫
R

{
|∂xun|2 + |∂xvn|2

}
dx+Gn(un, vn),

Gn(un, vn) =
1

2

∫
R

{
Πn(|un|) + Πn(|vn|)− Ξn(|un|)− Ξn(|vn|)

− 2

p+ 1
|Jnun|p+1|Jnvn|p+1

}
dx,

(2.13)

and

Πn(z) =
1

2

∫ |z|
0

ζn(s)ds, Ξn(z) =
1

2

∫ |z|
0

βn(s)ds. (2.14)

Step 2. We set

θn = sup{τ > 0 : ‖(un(t), vn(t))‖H1×H1 ≤ 2M on (−τ, τ)}
It is easily seen that

‖gj(u, v)− gj(w, z)‖L2×L2 ≤ C(M)‖(u, v)− (w, z)‖L2×L2 , j = 1, 2, (2.15)

for all (u, v), (w, z) ∈ H1(R)×H1(R) such that ‖(u, v)‖H1×H1 + ‖(w, z)‖H1×H1 ≤
M .

By (2.9), g1,n, g2,n satisfy (2.15) with C(M) independent on n. Hence, from
(2.11), we obtain

sup
n∈N
‖(∂tun, ∂tvn)‖L∞((−θn,θn),H−1(R)×H−1(R)) ≤ C(M). (2.16)

Using [9, Lemma 3.6] and (2.16), we conclude that

‖(un(t), vn(t))− (u0, v0)‖L2×L2 ≤ C(M)|t|1/2 for all t ∈ (−θn, θn) (2.17)

Applying (2.17), the conservation of energy and charge, and that

G′n = (gn + g1,n, gn + g2,n),

we obtain

‖(un(t), vn(t)‖2H1×H1 ≤ ‖(u0, v0)‖2L2×L2 + ‖(u′0, v′0)‖2L2

+ 2|Gn(un(t), vn(t))−Gn(u0, v0)|

≤ ‖(u0, v0)‖2H1×H1 + C(M)|t|1/2, t ∈ (−θn, θn).

We define T (M) by

C(M)T (M)1/2 = 2M2.



8 L. CELY EJDE-2023/76

Then

‖(un, vn)‖L∞((−T,T ),H1) < 2M for T = min{T (M), θn}.
Hence T (M) ≤ θn. Therefore

‖(un, vn)‖L∞((−T (M),T (M)),H1×H1) ≤ 2M, (2.18)

‖(∂tun, ∂tvn)‖L∞((−T (M),T (M)),H−1×H−1) ≤ C(M) (2.19)

Step 3. From (2.18), (2.19) and [9, Proposition 1.3.14], we conclude that there
exist (u(t), v(t)) in

L∞((−T (M), T (M)), H1 ×H1) ∩W 1,∞((−T (M), T (M)), H−1 ×H−1),

and a subsequence, which we still denote by (un, vn), such that

(un(t), vn(t)) ⇀
n→∞

u(t) in H1(R),

for all t ∈ [−T (M), T (M)],
Now since En(u0, v0)→ E(u0, v0) as n goes to +∞ and Πn ≥ 0, it follows that

‖un(t)‖2H1 + ‖vn(t)‖2H1 ≤ C + ‖Ξn(un(t))‖L1 + ‖Ξn(vn(t))‖L1

+ C‖un(t)vn(t)‖p+1
Lp+1

(2.20)

In addition, a simple calculation shows that there exists a constant C such that
(see [9, p. 296])

‖Ξn(un(t))‖L1 + ‖Ξn(vn(t))‖L1 ≤ 1

4
(‖un(t)‖2H1 + ‖vn(t)‖2H1)

+ C(‖un(t)‖2L2 + ‖vn(t)‖2L2).
(2.21)

Thus, combining (2.6), (2.20), and (2.21) we infer that (un, vn) is bounded in
L∞(R, H1(R) × H1(R)). In particular, we easily verify that (see [9, Step 2 in
Theorem 9.3.4])

un|un|p−1|vn|p+1 and vn|vn|p−1|un|p+1 are bounded in L∞(R, H−1(R)), (2.22)

gn(un) and gn(vn) are bounded in L∞(R, H−1(Ωk)) for every k ∈ R, (2.23)

where Ωk = {x ∈ R : |x| ≤ k}. It follows from (2.11), (2.22) and (2.23) that ∂tu
n
∣∣
Ωk

and ∂tv
n
∣∣
Ωk

are bounded in L∞(R, H−1(Ωk)). By the Sobolev’s embedding and

Arzela-Ascoli compactness criterion there exist (u, v) ∈ L∞(R, H1(R)×H1(R)) and
a subsequence, which we still denote by (un, vn), such that (see [9, Lemma 9.3.6])

(i) (un|Ωk , vn|Ωk) ∈W 1,∞(R, H−1(Ωk))×W 1,∞(R, H−1(Ωk)) for all k ∈ R.
(ii) (un(t), vn(t)) ⇀ (u, v) in H1(R)×H1(R) as n→ +∞ for every t ∈ R.

(iii) (un(x, t), vn(x, t)) ⇀ (u(x, t), v(x, t)) as n → +∞, for almost every x ∈ R
and for every t ∈ R.

Now, since gn(z) → z log |z|2 as n goes to +∞ and by properties (i)-(iii) above, it
is not hard to see that the limiting function (u, v) ∈ L∞(R,W (R)×W (R)) and it
is a weak solution to the equation (1.1) such that (u(0), v(0)) = (u0, v0). Moreover,
by weak lower semicontinuity, Fatou’s lemma and arguing in the same manner as
in [9, Step 3 in Theorem 9.3.4]) it is easy to see that ‖un(t)‖2L2 + ‖vn(t)‖2L2 =
‖u0‖2L2 + ‖v0‖2L2 and

E(u(t), v(t)) ≤ E(u0, v0) for every t ∈ R.



EJDE-2023/76 NONLINEAR SCHRÖDINGER SYSTEMS 9

Next we show uniqueness of the weak solution (u, v) ∈ L∞(R,W (R) × W (R))
to the equation (1.1). Assume the existence of two solutions (u, v) and (z, w) of
(1.1) with the same initial data (u0, v0) in the class L∞(R,W (R) ×W (R)). We
will prove that (u, v) ≡ (z, w). It suffices to show that u − z and v − w satisfy
‖u− z‖2L2 = ‖v−w‖2L2 = 0 for every t ∈ [0, T ], where T is a fixed positive number.
Indeed, on taking the difference of the two equations

i∂t(u− z) + ∂2
x(u− z) + u log |u|2 − z log |z|2 + u|u|p−1|v|p+1 − z|z|p−1|w|p+1 = 0

i∂t(v − w) + ∂2
x(v − w) + v log |v|2 − w log |w|2 + v|v|p−1|u|p+1 − w|w|p−1|z|p+1 = 0

and multiplying this equation by (i(u− z), i(v − w)) we infer that

− 1

2
∂t(‖u− z‖2L2 + ‖v − w‖2L2)

= Im

∫
R

(u log |u|2 − z log |z|2)(u− z)dx+ Im

∫
R

(v log |v|2 − w log |w|2)(v − w)

+ Im

∫
R

(u|u|p−1|v|p+1 − z|z|p−1|w|p+1)(u− z)dx

+ Im

∫
R

(v|v|p−1|u|p+1 − w|w|p−1|z|p+1)(v − w)dx.

Then from Lemma 2.3 we see that there exists a constant C > 0 such that

‖u(t)− z(t)‖2L2 + ‖v(t)− w(t)‖2L2 ≤ C
∫ t

0

{‖u(t)− z(t)‖2L2 + ‖v(t)− w(t)‖2L2} dt.

Therefore, uniqueness of solution follows by Gronwall’s inequality. Finally, the
conservation of energy, and the continuity of the solution (u, v) ∈ C(R,W (R) ×
W (R)) follow from the same kind of arguments as in [9, Step 4 of Theorem 9.3.3].
This completes of proof. �

3. Existence and stability of standing waves

In this section, we study the existence of minimizers of problem (1.5) and the
orbital stability of G(η, ζ). Thus, the aim is to prove Theorem 1.2 and Corollary
1.3. First, we need some preliminary lemmas.

Lemma 3.1. Let η, ζ > 0. Every minimizing sequence for J(η, ζ) is bounded in B.
Moreover, J(η, ζ) > −∞.

Proof. Let {(un, vn)} be a minimizing sequence for J(η, ζ), then we find that ‖un‖2L2

and ‖vn‖2L2 are bounded. Now, from the Gagliardo-Nirenberg inequality (see (2.6))
we obtain that

‖unvn‖p+1
Lp+1 ≤ C(‖un‖2L2 + ‖vn‖2L2)

p+2
2 (‖∂xun‖2L2 + ‖∂xvn‖2L2)p/2. (3.1)

Moreover, by logarithmic Sobolev inequality with α2 = π/2 we have∫
R
|un|2 log |un|2dx ≤

1

2
‖∂xun‖2L2 + (log ‖un‖2L2 − (1 + log

√
π/2))‖un‖2L2

≤ 1

2
‖∂xun‖2L2 + η log η + Cη,

(3.2)

and ∫
R
|vn|2 log |vn|2dx ≤

1

2
‖∂xvn‖2L2 + ζ log ζ + Cζ. (3.3)



10 L. CELY EJDE-2023/76

Combining (3.1), (3.2), and (3.3) we have

2E(un, vn) + η log η + ζ log ζ + C(η + ζ) + C(‖∂xun‖2L2 + ‖∂xvn‖2L2)p/2

≥ 1

2
(‖∂xun‖2L2 + ‖∂xvn‖2L2)

Since E(un, vn) is bounded and 1 ≤ p < 2, it follows that (un, vn) is bounded in
H1 ×H1. Moreover, by (2.1), (2.3), and (3.1) we have∫

R
{Φ(|un(x)|) + Φ(|vn(x)|)}dx

≤ 2 sup
n
E(un, vn) +

∫
R
{Ψ(|un(x)|) + Ψ(|vn(x)|)}dx+

2

p+ 1
|un|p+1|vn|p+1 ≤ C.

Thus, by Lemma 2.1, we have that ‖un‖2W (R) and ‖vn‖2W (R) are bounded, hence

‖(un, vn)‖B is bounded. Finally, if (u, v) ∈ B, ‖u‖2L2 = η, and ‖v‖2L2 = ζ, by (3.1),
(3.2), and (3.3), it follows that E(u, v) is bounded below, hence J(η, ζ) > −∞.
This completes the proof. �

For each minimizing sequence {(fn, gn)} of J(η, ζ), we define

γ = lim
r→∞

lim
n→∞

sup
y∈R

∫ y+r

y−r
(|fn|2 + |gn|2) dx.

Notice that γ satisfies 0 ≤ γ ≤ η+ζ. It follows from the concentration compactness
principle due to Lions [18] that there are three mutually exclusive cases:

(i) (Vanishing) γ = 0. This means that

lim
n→∞

sup
y∈R

∫ y+r

y−r
(|fn|2 + |gn|2) dx = 0, for every r ≥ 0.

(ii) (Dichotomy) 0 < γ < η + ζ, or
(iii) (Compactness) γ = η + ζ. That is, given ε > 0 there exist rε > 0 and a

sequence {yn} in R such that, for all n, we have∫ yn+rε

yn−rε
(|fn|2 + |gn|2) dx ≥ η + ζ − ε.

Let us study each case separately. First, we rule out the vanishing case (case (i)).
The following lemma can be proved by almost the same way as for [7, Lemma 3.2].
We omit the proof.

Lemma 3.2. Let {(fn, gn)} be a minimizing sequence for J(η, ζ). If p > 2, then
there are positive constants M = M(p, η) and C = C(p, ζ) such that ‖fn‖pLp ≥ M
and ‖gn‖pLp ≥ C for all n.

The following classical lemma is needed to rule out the case of vanishing. For a
proof, see [16, Lemma 3.9].

Lemma 3.3. Let {fn} be a bounded sequence in H1(R). If p > 2 and

lim
n→∞

sup
y∈R

∫ y+r

y−r
|fn(x)|2 dx = 0 (3.4)

for some r > 0, then limn→∞ ‖fn‖Lp = 0.

The following lemma states that γ = 0 does not occur.



EJDE-2023/76 NONLINEAR SCHRÖDINGER SYSTEMS 11

Lemma 3.4. If {(fn, gn)} is a minimizing sequence for J(η, ζ), then γ > 0.

Proof. If γ = 0 then there are exist r > 0 and a subsequence {(fnk , gnk)} such that

lim
n→∞

sup
y∈R

∫ y+r

y−r
(|fnk |2 + |gnk |2) dx = 0.

Since the sequences {fn} and {gn} are bounded in H1(R) by Lemma 3.1, then, by
Lemma 3.3, we infer that limn→∞ ‖fnk‖Lp = 0 and limn→∞ ‖gnk‖Lp = 0 for p > 2,
which contradicts Lemma 3.2. �

We now establish the following sequence of lemmas to rule out the dichotomy
case.

Lemma 3.5. If {(fn, gn)} is a minimizing sequence for J(η, ζ), then there exist
positive constants κ and δ such that, for n large enough,

(1) ‖∂xfn‖2L2 ≥ κ, if η > 0 and ζ ≥ 0.
(2) ‖∂xgn‖2L2 ≥ δ, if η ≥ 0 and ζ > 0.

Proof. Suppose (1) is false. Then there exists a subsequence {fnk} such that
limn→∞ ‖∂xfnk‖2L2 = 0. Since (fn) is bounded in H1(R) by Lemma 3.1, then
from Gagliardo-Nirenberg inequality we have that

‖fnk‖6L6 ≤ C‖∂xfnk‖2L2‖fnk‖4L2 → 0 if n→∞,
and this contradicts Lemma 3.2. Arguing as in the proof of (1), we obtain (2). This
completes the proof. �

The proof of the following lemma can be found in [7, Lemma 3.6].

Lemma 3.6. Let η > 0. Denote E1(f) = E(f, 0) and consider J1(η) given by

J1(η) = inf
{
E1(f) : f ∈W (R)textand‖f‖2L2 = η

}
. (3.5)

Let {fn} be a sequence in W (R) such that limn→∞ ‖fn‖2L2 = η and limn→∞E1(f) =
J1(η), then there exists a real number θ, a subsequence {fnk} of {fn} and a sequence
{yk} in R such that {eiθfnk(·+ yk)} converges strongly to φ

η
(x) in W (R), where

φ
η
(x) :=

( 1

π

)1/4
η1/2e−x

2/2, x ∈ R. (3.6)

In particular, ‖φ
η
‖2L2 = η and J1(η) = E1(φ

η
).

Recall that given a measurable function h : [0,∞)→ R, h∗ denotes the symmetric-
decreasing rearrangement of h,

h∗(x) =

∫ ∞
0

χ∗
{|h|>t}

(x) dt,

where χ∗
{|h|>t}

denotes the characteristic function of a ball of volume m({x : |h(x)| >
t}) centered at the origin. Here, m is the Lebesgue measure. The following lemma
shows that when f and g are replaced by |f∗| and |g∗| we have that E(|f∗|, |g∗|)
decreases.

Lemma 3.7. If (f, g) ∈ B, then (|f |∗, |g|∗) ∈ B and

E(|f |∗, |g|∗) ≤ E(|f |, |g|) ≤ E(f, g).

The proof of Lemma 3.7 can be found in [7, Lemma 3.7].



12 L. CELY EJDE-2023/76

Lemma 3.8. Let η, ζ ≥ 0 given, then there exist a minimizing sequence {(fn, gn)}
for J(η, ζ) such that the functions fn and gn belong to H1(R), are even, non-
increasing and non-negative for each n and x ≥ 0, and satisfy the condition ‖fn‖2L2 =
η, ‖gn‖2L2 = ζ.

The previous result can be proved in almost the same way as [7, Lemma 3.8].
We omit it here. The following lemma states the strict sub-additivity of J(η, ζ).
For a proof see [2, Lemma 2.10].

Lemma 3.9. Suppose that u, v : R→ [0,∞) are even, C∞ functions with compact
support in R, which are non-increasing on {x : x ≥ 0}. Let a1 and a2 be numbers
such that u(x+ a1) and v(x+ a2) have disjoint supports, and define

w(x) = u(x+ a1) + v(x+ a2).

Let w? : R→ R be the symmetric decreasing rearrangement of w. Then the distri-
butional derivative (w?)′ of w? is in L2 and satisfies

‖(w?)′‖2 ≤ ‖w′‖2 − 3

4
min{‖u′‖2, ‖v′‖2}.

The following two lemmas are central to show that dichotomy cannot occur.

Lemma 3.10. For each minimizing sequence {(fn, gn)} of J(η, ζ), there exists a
(µ1, µ2) ∈ [0, η]× [0, ζ] such that γ = η + ζ and

J(µ1, µ2) + J(η − µ1, ζ − µ2) ≤ J(η, ζ). (3.7)

Proof. Let φ ∈ C∞0 [−2, 2] be such that φ is identically 1 on [−1, 1], and let ψ ∈
C∞(R) be such that φ2 +ψ2 = 1 on R. Define φr(x) = φ(xr ) and ψr(x) = ψ(xr ), for
r > 0. From the definition of γ it follows that for given ε > 0 and for all sufficiently
large r we obtain that

γ − ε < lim
n→∞

sup
y∈R

∫ y+r

y−r
(|fn|2 + |gn|2) dx ≤ lim

n→∞
sup
y∈R

∫ y+2r

y−2r

(|fn|2 + |gn|2) dx ≤ γ.

We may assume that 1/r < ε, thus we may choose a positive integer N large enough
so that for all n ≥ N we have

γ − ε < sup
y∈R

∫ y+r

y−r
(|fn|2 + |gn|2) dx ≤ sup

y∈R

∫ y+2r

y−2r

(|fn|2 + |gn|2) dx < γ + ε.

Hence, for all n ≥ N , we can find yn such that∫ yn+r

yn−r
(|fn|2 + |gn|2)dx > γ − ε and

∫ yn+2r

yn−2r

(|fn|2 + |gn|2)dx < γ + ε. (3.8)

We define (
hn(x), ln(x)

)
= (φr(x− yn)fn(x), φr(x− yn)gn(x))(

h̃n(x), l̃n(x)
)

= (ψr(x− yn)fn(x), ψr(x− yn)gn(x)).

It follows from Lemma 3.1 that (hn), (ln), (h̃n), and (l̃n) are bounded sequences
in L2(R). Then, passing to a subsequence if necessary, we have ‖hn‖2L2 → µ1,

‖ln‖2L2 → µ2, ‖h̃n‖2L2 → η − µ1, and ‖l̃n‖2L2 → ζ − µ2 as n→∞, where (µ1, µ2) ∈
[0, η]× [0, ζ]. Also,

µ1 + µ2 = lim
n→∞

∫
R

(|hn|2 + |ln|2)dx = lim
n→∞

∫
R
φ2
r(|fn|2 + |gn|2)dx.



EJDE-2023/76 NONLINEAR SCHRÖDINGER SYSTEMS 13

Thus by (3.8) we have |(µ1 + µ2)− γ| < ε. Moreover, for all n, we infer that

E(hn, ln) + E(h̃n, l̃n) ≤ E(fn, gn) + Cε. (3.9)

Indeed, first notice that

(i) ‖∂xφr‖L∞ = 1
r‖∂xφ‖L∞ ≤ Cε;

(ii) Since φr is identically 1 on |x − yn| ≤ r and φr vanishes on |x − yn| ≥ 2r,
then |φr|2 log |φr|2 vanishes on both |x − yn| ≤ r and |x − yn| ≥ 2r. Next
by (3.8) we see that∫

R
(|fn|2 + |gn|2)|φr|2 log |φr|2dx ≤ 2e−1

∫
r≤|x−yn|≤2r

(|fn|2 + |gn|2)dx ≤ Cε;

(iii) Also, again using (3.8) and (2.6), we have inferred that∫
R

(
(φ2
r − φ2(p+1)

r )|fn|p+1|gn|p+1
)
dx ≤ C

∫
r≤|x−yn|≤2r

(|fn|2 + |gn|2)dx ≤ Cε.

And so, to prove (3.9), we write

2E(hn, ln) =

∫
R
φ2
r

(
|∂xfn|2 + |∂xgn|2 − |fn|2 log |fn|2 − |gn|2 log |gn|2

− 2

p+ 1
|fn|p+1|gn|p+1

)
dx+

∫
R

2φr∂xφr
(
Refn∂xfn + Regn∂xgn

)
dx

+

∫
R

(|fn|2 + |gn|2)
(
(∂xφr)

2 − |φr|2 log |φr|2
)
dx

+
2

p+ 1

∫
R

(
(φ2
r − φ2(p+1)

r )|fn|p+1|gn|p+1
)
dx

≤
∫
R
φ2
r

(
|∂xfn|2 + |∂xgn|2 − |fn|2 log |fn|2 − |gn|2 log |gn|2

− 2

p+ 1
|fn|p+1|gn|p+1

)
dx+ Cε.

In the same way, we obtain

2E(h̃n, l̃n) ≤
∫
R
ψ2
r

(
|∂xfn|2 + |∂xgn|2 − |fn|2 log |fn|2 − |gn|2 log |gn|2

− 2

p+ 1
|fn|p+1|gn|p+1

)
dx+ Cε.

Then (3.9) follows by combining the two estimates and using φ2
r +ψ2

r = 1. Now we
assume that µ1, µ2, η − µ1 and ζ − µ2 are all positive. We define

αn =

√
µ1

‖hn‖L2

, βn =

√
µ2

‖ln‖L2

, νn =

√
η − µ1

‖h̃n‖L2

, ϑn =

√
ζ − µ2

‖l̃n‖L2

.

Then αn, βn νn, and ϑn tend to 1 as n→∞ and

‖αnhn‖2L2 = µ1, ‖βnln‖2L2 = µ2, ‖νnh̃n‖2L2 = η − µ1, ‖ϑn l̃n‖ = ζ − µ2.

Thus
lim inf
n→∞

{
E(hn, ln) + E(h̃n, l̃n)

}
≥ J(µ1, µ2) + J(η − µ1, ζ − µ2).

On the other hand, if µ1 = 0 and µ2 > 0, then ‖hn‖L2 → 0 and from Gagliardo-
Nirenberg inequality we infer that

‖hn‖p+1
Lp+1 ≤ C‖∂xhn‖

p−1
2

L2 ‖hn‖
p+1

2

L2 → 0.



14 L. CELY EJDE-2023/76

Thus
∫
R |hn|

p+1|ln|p+1dx→ 0. By (2.3),
∫
R Ψ(|hn|)dx→ 0. Since Φ ≥ 0 we have

lim
n→∞

E(hn, ln) = lim
n→∞

∫
R
(|∂xhn|2 + |∂xln|2 + Φ(|hn|)− |ln|2 log |ln|2)dx

≥ lim inf
n→∞

(|∂xln|2 − |ln|2 log |ln|2)

≥ J(0, µ2),

and similar estimates hold if µ2, η − µ1 or ζ − µ2 are zero. Thus, in all cases we
have that the limit inferior as n→∞ of the left-hand side of (3.9) is greater than
or equal to J(µ1, µ2) + J(η − µ1, ζ − µ2). We now take the limit inferior of the
left-hand side and the limit of the right-hand side of (3.9) as n→∞ to obtain

J(µ1, µ2) + J(η − µ1, ζ − µ2) ≤ J(η, ζ) + Cε,

hence J(µ1, µ2) + J(η − µ1, ζ − µ2) ≤ J(η, ζ), since ε is arbitrary. �

Lemma 3.11. For any η1, ζ1, η2, ζ2 ≥ 0 such that (η1, ζ1), (η2, ζ2) 6= (0, 0), we have

J(η1 + ζ2, η1 + ζ2) < J(η1, ζ1) + J(η2, ζ2).

Proof. Let (f
(k)
n , g

(k)
n ) be the minimizing sequences of J(ηk, ζk) given by Lemma

3.8, with k = 1, 2. Then, for each n, we can choose xn such that both f
(1)
n and

hn(x) := f
(2)
n (x + xn) as g

(1)
n and h̃n(x) := g

(2)
n (x + xn) are disjointly supported.

We define
un := (f1

n + hn)∗ and vn := (g1
n + h̃n)∗.

Then ‖un‖2L2 = η1 + η2 and ‖vn‖2L2 = ζ1 + ζ2, so

J(η1 + ζ2, η1 + ζ2) ≤ E(un, vn). (3.10)

Now, from Lemma 3.9 we have∫
R
(∂xu

2
n + ∂xv

2
n)dx

≤
∫
R
{(∂xf (1)

n )2 + (∂xhn)2 + (∂xg
(1)
n )2 + (∂xh̃n)2}dx− Tn,

(3.11)

where

Tn =
3

4
(min{‖∂xf (1)

n ‖2L2 , ‖∂xhn‖2L2}+ min{‖∂xg(1)
n ‖2L2 , ‖∂xh̃n‖2L2}). (3.12)

Furthermore, from [17, Theorem 3.4] and the properties of rearrangements (see
Lemma 3.7), we have∫

R
|un|p+1|vn|p+1dx ≥

∫
R
|f (1)
n |p+1|g(1)

n |p+1dx+

∫
R
|hn|p+1|h̃n|p+1dx∫

R
|un|2 log |un|2dx =

∫
R
|f (1)
n |2 log |f (1)

n |2dx+

∫
R
|hn|2 log |hn|2dx∫

R
|vn|2 log |vn|2dx =

∫
R
|g(1)
n |2 log |g(1)

n |2dx+

∫
R
|h̃n|2 log |h̃n|2dx,

hence, combining this with (3.10) and (3.11), we have

J(η1 + η2, ζ1 + ζ2) ≤ E(un, vn) ≤ E(f (1)
n , g(1)

n ) + E(hn, h̃n)− Tn
for every n, and taking the limit superior on the right-hand side, we obtain

J(η1 + η2, ζ1 + ζ2) ≤ J(η1, ζ1) + J(η2, ζ2)− lim inf
n→∞

Tn. (3.13)



EJDE-2023/76 NONLINEAR SCHRÖDINGER SYSTEMS 15

Since ζ1 + ζ2 > 0, then either ζ1 and ζ2 are both positive, or ζ1 = 0 and ζ2 > 0 or
ζ1 > 0 and ζ2 = 0. As noted in [2, Lemma 2.12], it is sufficient to consider three
cases:

(1) Suppose that ζ1 > 0 and ζ2 > 0. By Lemma 3.5 there exist κ1 and κ2

such that ‖∂xg(1)
n ‖2L2 ≥ κ1 and ‖∂xh̃n‖2L2 ≥ κ2, for all sufficiently large n. Let

κ = min{κ1, κ2}, then, it follows from (3.12) and (3.13) that Tn ≥ 3κ/4 for all
sufficiently large n, and

J(η1 + η2, ζ1 + ζ2) ≤ J(η1, ζ1) + J(η2, ζ2)− 3

4
κ < J(η1, ζ1) + J(η2, ζ2).

(2) Suppose that ζ1 = 0, ζ2 > 0 and η2 > 0. Since η1 + ζ1 > 0, η1 > 0, then η1 > 0.

By Lemma 3.5 there exist δ1 and δ2 such that ‖∂xf (1)
n ‖2L2 ≥ δ1 and ‖∂xhn‖2L2 ≥ δ2,

for all sufficiently large n. Let δ = min{δ1, δ2}, then, again from (3.12) and (3.13),
we obtain that Tn ≥ 3δ/4 for all sufficiently large n, and

J(η1 + η2, ζ1 + ζ2) ≤ J(η1, ζ1) + J(η2, ζ2)− 3

4
δ < J(η1, ζ1) + J(η2, ζ2).

(3) Suppose that ζ1 = 0, ζ2 > 0 and η2 = 0. By Lemma 3.6 we have

J(η1, 0) = inf
{1

2

∫
R

(
|∂xu|+ |u|2 − |u|2 log |u|2

)
dx : u ∈W (R), ‖u‖2L2 = η1 > 0

}
,

where the minimum is achieved at ϕ
η1

(x) = ( 1
π )1/4(η1)1/2e

−x2

2 . Also,

J(0, ζ2) = inf
{1

2

∫
R

(|∂xv|+ |v|2 − |v|2 log |v|2)dx : v ∈W (R), ‖v‖2L2 = ζ2 > 0
}
,

where the minimum is achieved at ϕ
ζ2

(x) = ( 1
π )1/4(ζ2)1/2e

−x2

2 . Therefore,

J(η1 + η2, ζ1 + ζ2) ≤ E(ϕη1
, ϕ

ζ2
) +Q(ϕ

η1
, ϕ

ζ2
)

=
1

2

∫
R

(
|∂xϕη1

|2 + |ϕ
η1
|2 − |ϕ

η1
|2 log |ϕ

η1
|2
)
dx

+
1

2

∫
R

(
|∂xϕζ2 |

2 + |ϕ
ζ2
|2 − |ϕ

ζ2
|2 log |ϕ

ζ2
|2
)
dx

− 2

p+ 1

∫
R
|ϕ

η1
|p+1|ϕ

ζ2
|p+1dx

= J(η1, 0) + J(0, ζ2)− 2

p+ 1

∫
R
|ϕη1
|p+1|ϕ

ζ2
|p+1dx

< J(η1, 0) + J(0, ζ2).

The proof is complete. �

We are now ready to prove that dichotomy of minimizing sequences does not
occur.

Lemma 3.12. Let η > 0 and ζ > 0. Then for any minimizing sequence {(fn, gn)}
of J(η, ζ), we have γ = η + ζ.

Proof. Suppose by way of contradiction that γ ∈ (0, η+ζ). Let η1, ζ2 be definite as
in Lemma 3.10, and let η2 = η − η1 and ζ2 = ζ − ζ2. Then η2 + ζ2 = η + ζ − γ > 0



16 L. CELY EJDE-2023/76

and η1 + ζ1 = γ > 0. Since η1 + η2 = η > 0 and ζ1 + ζ2 = ζ > 0, in view of Lemma
3.11, we have

J(η1 + η2, ζ1 + ζ2) < J(η1, ζ1) + J(η2, ζ2),

but this contradicts (3.7). �

Thus, we eliminated the cases γ = 0 and 0 < γ < η+ ζ, it follows that γ = η+ ζ.
Then we can prove that, up to translation, any minimizing sequence is precompact
in B and problem (1.5) has at least one minimizer.

Lemma 3.13. Let η > 0 and ζ > 0. Then, for any minimizing sequence {(fn, gn)}
of J(η, ζ), there exists a sequence of real numbers (yn) such that, by passing to
subsequence if necessary, {fn(· + yn), gn(· + yn)} converges strongly in B to some
minimizer (f, g) of J(η, ζ).

Proof. By Lions’ concentration compactness lemma [18, 19], since γ = η + ζ we
have that, for every k ∈ N, there exists λk such that∫ λk

−λk
(|fn(x+ yn)|2 + |gn(x+ yn)|2) dx > η + ζ − 1

k
,

for all sufficiently large n. Hence, in view of the embedding H1
loc(R) ↪→ L2

loc(R),
it follows that some further subsequence of {fn(· + yn), gn(· + yn)} converges in
L2[−λk, λk]-norm to a limit (f, g) ∈ L2[−λk, λk]× L2[−λk, λk]. Moreover,∫ λk

−λk
(|f |2 + |g|2) dx > η + ζ − 1

k
.

Then we apply Cantor’s diagonalization process, and we obtain that {fn(·+yn), gn(·+
yn)} converges in L2(R)-norm to a limit (f, g) ∈ L2(R)× L2(R) and∫

R
(|f |2 + |g|2) dx = η + ζ.

Thus, (fn(·+ yn), gn(·+ yn)) ⇀ (f, g) weakly in B, since B is reflexive. Now, since
the sequences {fn(·+ yn)} and {gn(·+ yn)} are bounded in H1(R), from (2.3) we
have

lim
n→∞

∫
R

Ψ(|fn(x+ yn)|) dx =

∫
R

Ψ(|f(x)|) dx, (3.14)

lim
n→∞

∫
R

Ψ(|gn(x+ yn)|) dx =

∫
R

Ψ(|g(x)|) dx. (3.15)

Also we have∫
R
|∂xf |2 dx+

∫
R

Φ(|f |) dx

≤ lim inf
n→∞

{∫
R
|∂xfn(x+ yn)|2 dx+

∫
R

Φ(|fn(x+ yn)|) dx
}
,

(3.16)

∫
R
|∂xg|2 dx+

∫
R

Φ(|g|) dx

≤ lim inf
n→∞

{∫
R
|∂xgn(x+ yn)|2 dx+

∫
R

Φ(|gn(x+ yn)|) dx
}
,

(3.17)



EJDE-2023/76 NONLINEAR SCHRÖDINGER SYSTEMS 17

because the functional h 7→
∫
R Φ(|h|)dx, Φ(|h|) = Ψ(|h|)−|h|2 log |h|2, is convex and

continuous on W (R), thus weakly l.s.c on W (R). Then, by (3.14), (3.15), (3.16),
and (3.17), we have

E(f, g) ≤ lim inf
n→∞

E(fn, gn) = J(η, ζ),

hence E(f, g) = J(η, ζ) and (f, g) ∈ G(η, ζ). Therefore,

lim
n→∞

{∫
R

(|∂xfn(x+ yn)|2 + |∂xgn(x+ yn)|2) + Φ(|fn(x+ yn)|)

+ Φ(|gn(x+ yn)|)dx
}

=

∫
R

(
|∂xf |2 + |∂xg|2

)
+ Φ(|f |) + Φ(|g|)dx.

Hence and in view of [10, Lemma 2.4.4] we have

lim
n→∞

∫
R
|∂xfn(x+ yn)|2 dx =

∫
R
|∂xf |2dx,

lim
n→∞

∫
R
|∂xgn(x+ yn)|2 dx =

∫
R
|∂xg|2dx,

lim
n→∞

∫
R

Φ(|fn(x+ yn)|)dx =

∫
R

Φ(|f |)dx,

lim
n→∞

∫
R

Φ(|gn(x+ yn)|)dx =

∫
R

Φ(|g|)dx.

Hence and in view of Lemma 2.1 we have fn → f and gn → g in LΦ(R). Thus,
by definition of the B-norm, we infer that {fn(·+ yn), gn(·+ yn)} converges in the
norm of B. That is, (fn, gn)→ (f, g) strongly in B as n→∞. �

Based on the above lemmas we are ready for the following proof.

Proof of Theorem 1.2. (i) is an immediate consequence of Lemma 3.13.
(ii) By the Lagrange multiplier principle, if (f, g) ∈ G(γ, µ), there exist real

numbers ρ and ϑ such that

E′(f, g) = ρf + ϑg,

where the prime denotes the Fréchet derivative. Since

E′(f, g) =

[
−∂2

xf − f log |f |2 − f |f |p−1|g|p+1

−∂2
xg − g log |g|2 − g|g|p−1|f |p+1

]
,

the equations

−∂2
xf + ρf = f log |f |2 + f |f |p−1|g|p+1

−∂2
xg − ϑg = g log |g|2 + g|g|p−1|f |p+1

(3.18)

hold at least in the sense of distributions. By a standard bootstrap argument, we
see that f and g are in C2(R) (see [9, Chapter 8]). Thus, we may write

f(x) = eiθ(x)φ(x), g(x) = eiω(x)ϕ(x),

where θ, ω, φ, ϕ ∈ C2(R) and φ, ϕ ≥ 0. It remains to prove that both θ and ω
are constants. By Lemma 3.7 (φ, ϕ) is a minimizer for J(η, ζ), then (φ, ϕ) satisfies



18 L. CELY EJDE-2023/76

(3.18). On the other hand, multiplying the first equation by f and the second by g
in (3.18) and integrating by parts over R we have

ρ = −1

η

∫
R
(|∂xf |2 − |f |2 log |f |2 − |f |p+1|g|p+1)dx

ϑ = −1

ζ

∫
R

(|∂xg|2 − |g|2 log |f |2 − |f |p+1|g|p+1)dx.

Hence, and E(f, g) = E(φ, ϕ), it follows that∫
R

(|∂xφ|2 − |φ|2 log |φ|2 − |φ|p+1|g|p+1)dx

=

∫
R

(|∂xf |2 − |f |2 log |f |2 − |f |p+1|g|p+1)dx.

(3.19)

Then computing the second derivative of f we have

∂2
xf = eiθ(x)

(
ρφ− |φ|2 log |φ|2 − |φ|p+1|g|p+1

)
− (∂xθ)

2φ+ 2i∂xθ∂xφ+ iφ∂2
xθ,

(3.20)

and by the first equality in (3.18) we infer that

∂2
xf = ρeiθφ− eiθ|φ|2 log |φ|2 − eiθ|φ|p+1|g|p+1. (3.21)

Thus by (3.20) and (3.21) we have

(∂xθ)
2φ+ 2i∂xθ∂xφ+ iφ∂2

xθ = 0, for all x ∈ R. (3.22)

Similarly, we obtain that

(∂xω)2ϕ+ 2i∂xω∂xϕ+ iϕ∂2
xω = 0, for all x ∈ R. (3.23)

Thus since θ, ω ∈ C2(R), then, taking the real part of (3.22) and (3.23), we have
∂xθ(x) = 0 and ∂xω(x) = 0, Hence θ and ω are constant, say θ = θ0 and ω = ω0,
thus f(x) = eiθ0φ(x) and g(x) = eiω0ϕ(x) for all x ∈ R. Finally, defining τ(s) =
ρs− s log s2, we have

∂2
xφ = ρφ− φ log φ2 − φp+1|g|p+1 ≤ τ(φ) on R.

Notice that τ is positive, continues, non-decreasing near zero, τ(0) = 0 and τ(e
ρ
2 ) =

0. Thus in view of [24, Theorem 1] we have φ = 0 or φ > 0. But ‖f‖2L2 = η > 0,
then φ > 0. Similarly, we obtain that ϕ > 0 for all x ∈ R.

(iii) It is trivial from the definition of true two-parameter family. This completes
of proof. �

Proof of Corollary 1.3. Let us suppose that G(η, ζ) is not B-stable. For each k ∈ N
there exist initial data (f0,k, g0,k) and (tk) ⊂ [0,+∞) such that for some ε > 0 and
all k we have

inf{‖(f0,k, g0,k)− (ϕ, φ)‖B : (ϕ, φ) ∈ G(η, ζ)} < 1

k

and

inf{‖(fk(tk), gk(tk))− (ϕ, φ)‖B : (ϕ, φ) ∈ G(η, ζ)} ≥ ε, (3.24)

where (fk(t), gk(t)) denotes the solution of (1.1) with initial data (f0,k, g0,k). Since

f0,k
k→ ϕ and g0,k

k→ φ in B and Q(ϕ, φ) = η + ζ, we have

lim
k→∞

Q(f0,k, g0,k) = η + ζ and lim
k→∞

E(f0,k, g0,k) = J(η, ζ). (3.25)



EJDE-2023/76 NONLINEAR SCHRÖDINGER SYSTEMS 19

Since E and Q are independent of t, by (3.25) we have

lim
k→∞

Q(fk(tk), gk(tk)) = η + ζ and lim
k→∞

E(fk(tk), gk(tk)) = J(η, ζ).

Hence, {(fk(tk), gk(tk))} is a minimizing sequence for J(η, ζ) and by Theorem 1.2,
up to a subsequence, there exist a sequence {yk} in R and functions (ν, ψ) ∈ G(η, ζ)
such that

lim
k→∞

‖(fk(·+ yk, tk), gk(·+ yk, tk))− (ν, ψ)‖B = 0

Since (ν(· − yk), ψ(· − yk)) ∈ G, we have

inf{‖(fk(tk), gk(tk))− (ϕ, φ) : (ϕ, φ) ∈ G(η, ζ)‖B} < ε

for all sufficiently large k, which is a contradiction with (3.24). This completes the
proof. �

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Liliana Cely

Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão, 1010,
São Paulo-SP, 05508-090, Brazil

Email address: mlcelyp@unal.edu.co


	1. Introduction
	2. Existence results
	3. Existence and stability of standing waves
	References