Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 06, pp. 1–18.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

STATIONARY QUANTUM ZAKHAROV SYSTEMS INVOLVING

A HIGHER COMPETING PERTURBATION

SHUAI YAO, JUNTAO SUN, TSUNG-FANG WU

Abstract. We consider the stationary quantum Zakharov system with a
higher competing perturbation

∆2u−∆u+ λV (x)u = K(x)uφ− µ|u|p−2u in R3,

−∆φ+ φ = K(x)u2 in R3,

where λ > 0, µ > 0, p > 4 and functions V and K are both nonnegative. Such

problem can not be studied via the common arguments in variational meth-
ods, since Palais-Smale sequences may not be bounded. Using a constraint

approach proposed by us recently, we prove the existence, multiplicity and
concentration of nontrivial solutions for the above problem.

1. Introduction

Our starting point is the quantum Zakharov system

i∂tE + ∆E − ε2∆2E = nE, (t, x) ∈ R× RN ,
∂2
t n−∆n+ ε2∆2n = ∆|E|2,

(1.1)

where N = 1, 2, 3, the dimensionless quantum coefficient 0 < ε ≤ 1, the complex
valued function E = E(t, x) is the envelope electric field and the real valued function
n = n(t, x) is the plasma density fluctuation. Such system has been introduced by
Garcia et al. [8] and Haas-Shukla [11] as a model describing the nonlinear interaction
between high-frequency quantum Langmuir waves and low-frequency quantum ion-
acoustic waves. For more physical meaning, we refer the reader to [10] and the
references therein.

In recent years, many researches have studied system(1.1), but they concern
mainly the well-posedness of initial value problems, see for example [3, 4, 7, 9, 12].
More precisely, when N = 1, Jiang-Lin-Shao [12] proved the local well-posedness
of system (1.1) with inital value (E0, n0, ∂tn0) ∈ Hk(R) × H l(R) × H l−2(R) pro-
vided that |k| − 3

2 < l < min{k + 3
2 , 2k + 3

2} and k > − 3
4 . Chen-Fang-Wang [3]

obtained the global well-posedness of system (1.1) with inital value (E0, n0, ∂tn0) ∈
L2(R) × H l(R) × H l−2(R) provided that −3/2 ≤ l ≤ 3/2. When N = 1, 2, 3,
Guo-Zhang-Guo [9] proved the global well-posedness of system (1.1) with initial
value (E0, n0, ∂tn0) ∈ Hk(RN ) × Hk−1(RN ) × Hk−3(RN ) with k ≥ 2. Moreover,

2010 Mathematics Subject Classification. 35J35, 35B38.

Key words and phrases. Quantum Zakharov system; variational methods; multiple solutions.
c©2020 Texas State University.

Submitted July 21, 2019. Published January 10, 2020.

1



2 S. YAO, J. SUN, T.-F. WU EJDE-2020/06

the classical limit behavior of system (1.1) was studied as the quantum parameter
ε→ 0.

If we look for the stationary solution and static solution in the form of

E(t, x) = eiωtu(x) and n(t, x) = φ(x),

then system (1.1) is deduced from the elliptic system:

−ε2∆2u+ ∆u− ωu = uφ in RN ,

ε2∆φ− φ = u2 in RN .
(1.2)

Recently, Fang-Segata-Wu [6] studied the existence of ground state solution for
system (1.2) with 0 < ε ≤ 1 and ω > 0. In addition, the existence of bound state
radial solution was obtained when ε > 0 is sufficiently small and ω > 0. Later,
in [17] the authors considered a class of quantum Zakharov systems with a local
perturbation, i.e.

∆2u−∆u+ λV (x)u = uφ− µf(x)|u|p−2u in R3,

−∆φ+ φ = u2 in R3,
(1.3)

where the parameters λ > 0, µ ∈ R and the potential V (x) satisfies the following
assumptions:

(A1) V ∈ C(R3,R) with V (x) ≥ 0 in R3 and there exists b > 0 such that
|{V < b}| is the finite, where | · | is the Lebesgue measure;

(A2) Ω = int{x ∈ R3 | V (x) = 0} is nonempty and has smooth boundary with
Ω = {x ∈ R3 : V (x) = 0}.

By using the Nehari manifold method, for λ sufficiently large, in [17] we con-
cluded the following results:

(i) when 1 < p < 2 and −µ1 < µ < 0, at least two nontrivial solutions exists
if f ∈ L2/(2−p)(R3);

(ii) when p = 2 and −µ2 < µ < 0, or p > 2 and µ < 0, or µ = 0, or 1 < p < 4
and µ > 0, or p = 4 and 0 < µ < µ3, a nontrivial ground state solution is
permitted if f ∈ L2/(2−p)(R3) for 1 < p < 2 and f ∈ L∞(R3) for p ≥ 2.

We notice that when µ > 0 and p > 4 of system (1.3)) has not been studied in [17],
since the competing effect of the nonlocal term with the perturbation gives rise
to methodological difficulties. Specifically, the common arguments in variational
methods, such as mountain pass theorem, can not be applied because Palais-Smale
sequences may not be bounded. Moreover, the Nehari manifold method does not
work as well, since the energy functional is not bounded below on it.

Motivated by the analysis above, in this paper we are interested in studying the
qualitative properties of nontrivial solutions in the case µ > 0 and p > 4, including
the existence, multiplicity and concentration. Having a little difference with system
(1.3), we consider the problem

∆2u−∆u+ λV (x)u = K(x)uφ− µ|u|p−2u in R3,

−∆φ+ φ = K(x)u2 in R3,
(1.4)

where λ > 0, µ > 0, p > 4 and V (x) satisfies conditions (A1) and (A2), and
K ∈ L∞(R3) ∪ L2p/(p−4)(R3) with K(x) ≥ 0 in R3.



EJDE-2020/06 STATIONARY QUANTUM ZAKHAROV SYSTEMS 3

As in [17], system (1.4) can be transformed into the following nonlinear bihar-
monic equation with a nonlocal term,

∆2u−∆u+ λV (x)u = K(x)uφK,u − µ|u|p−2u in R3, (1.5)

where

φK,u(x) =
1

4π

∫
R3

K(y)u2(y)

|x− y| exp(|x− y|)
dy.

Equation (1.5) is variational and its solutions are the critical points of the functional
given by

Iλ,µ(u) =
1

2
‖u‖2λ −

1

4

∫
R3

K(x)φK,uu
2dx+

µ

p

∫
R3

|u|pdx,

where ‖u‖λ =
∫
R3(|∆u|2 + |∇u|2 + λV (x)u2)dx. The functional Iλ,µ is of class C1

in Xλ (see Section 2) whose Fréchet derivative is given by

〈I ′λ,µ(u), v〉 =

∫
R3

(∆u∆v +∇u∇v + λV (x)uv)dx−
∫
R3

K(x)φK,uuv dx

+ µ

∫
R3

|u|p−2uv dx

for any v ∈ H2(R3). Hence, if u ∈ Xλ is a critical point of Iλ,µ, then (u, φK,u) is a
solution of system (1.4).

Very recently, we proposed a novel constraint approach to find critical points in
the study of Schrödinger-Poisson systems [15, 16] and Kirchhoff type problems [14].
Such approach can effectively solve the difficulties concerned above. In this paper,
we shall further develop it to investigate system (1.4) with µ > 0 and p > 4. To be
specific, by introducing the filtration of the Nehari manifold as follows

Nλ,µ(c) = {u ∈ Nλ,µ : Iλ,µ(u) < c} for some c > 0,

where Nλ,µ is the Nehari manifold, we prove that Nλ,µ(c) can be decomposed as

Nλ,µ(c) = N
(1)
λ,µ(c) ∪N

(2)
λ,µ(c),

where

N
(1)
λ,µ(c) = {u ∈ Nλ,µ(c) : ‖u‖λ < D}, N

(2)
λ,µ(c) = {u ∈ Nλ,µ(c) : ‖u‖λ > D}

for 0 < D < D, in which each local minimizer of the functional Iλ,µ is a critical

point of Iλ,µ in H2(R3). In consideration of the boundedness of N
(1)
λ,µ(c), we can

minimize the functional Iλ,µ on N
(1)
λ,µ(c), where Iλ,µ is bounded below, to find a

critical point. Furthermore, if we can further prove that N
(2)
λ (c) is bounded and

that Iλ,µ is bounded below on N
(2)
λ (c), then two critical points can be found by

minimizing Iλ,µ on both N
(1)
λ,µ(c) and N

(2)
λ,µ(c).

Before stating our results, we introduce some notation. Denote by S∞ is the
best Sobolev constant for the embedding of H2(R3) in L∞(R3). Let A > 0 be the
sharp constant of Gagliardo-Nirenberg inequality and α > 0 be the least energy of
the limiting equation (see (2.10) below). Let

µ∗ =
2

(p− 4)(1 + Ā16/3|{V < b}|4/3)p/2
[S2
∞(p− 4)

8(p− 2)α

](p−2)/2
> 0.

We summarize our main results as follows.



4 S. YAO, J. SUN, T.-F. WU EJDE-2020/06

Theorem 1.1. Suppose that p > 4,K ∈ L∞(R3) with K(x) ≥ 0 and conditions
(A1) and (A2) hold. Then there exists a number Λ∗ > 0 such that for every
λ ≥ Λ∗ and 0 < µ < µ∗, system (1.4) admits at least one nontrivial solution
(u−λ,µ, φK,u−λ,µ

) ∈ H2(R3)×H1(R3) which satisfies

‖u−λ,µ‖λ < 2
[α(p− 2)

p− 4

]1/2
and 0 < Iλ,µ(u−λ,µ) <

(p− 2)2

p(p− 4)
α.

Theorem 1.2. Assume that p > 4,K ∈ L2p/(p−4)(R3) with K(x) ≥ 0 and con-
ditions (A1) and (A2) hold. Then there exists a number Λ ≥ Λ∗ such that for
each λ > Λ and 0 < µ < µ∗, system (1.4) admits at least two nontrivial solutions
(u±λ,µ, φK,u±λ,µ

) ∈ H2(R3)×H1(R3) which satisfy

‖u−λ,µ‖λ < 2
[α(p− 2)

p− 4

]1/2
< ‖u+

λ,µ‖λ,

Iλ,µ(u+
λ,µ) < 0 < Iλ,µ(u−λ,µ) <

(p− 2)2

p(p− 4)
α.

In particular, (u+
λ,µ, φK,u+

λ,µ
) is a ground state solution.

Theorem 1.3. Suppose that (u±λ,µ, φK,u±λ,µ
) are the nontrivial solutions of (1.4)

obtained by Theorem 1.2. Then (u±λ,µ, φK,u±λ,µ
)→ (u±∞, φK,u±∞) in H2(R3)×H1(R3)

as λ→∞ where u±∞ ∈ H2
0 (Ω) are nontrivial weak solutions of the Dirichlet problem

∆2u−∆u =
1

4π
K(x)

(∫
Ω

K(y)u2(y)

|x− y| exp(|x− y|)
dy
)
u− µ|u|p−2u in Ω,

u =
∂u

∂n
= 0 on ∂Ω,

(1.6)

Remark 1.4. In [17], when 1 < p < 2 and µ < 0, we obtained the existence of two
nontrivial solutions: one is in the neighborhood of the origin whose energy level is
negative and the other’s energy level is positive. In fact, such case is very similar
to the one of concave-convex term. Theorem 1.2 shows that when p > 4 and µ > 0,
two nontrivial solutions can also be found. However, the solution with negative
energy level is away from the origin, which is distinguished from the one in [17].

The remainder of this paper is organized as follows. After presenting some
preliminary results in section 2, we prove Theorems 1.1 and 1.2 in sections 3 and
4, respectively. Finally, we explore the concentration of solutions in the section 5.

2. Preliminaries

Let

X =
{
H2(R3) :

∫
R3

(|∆u|2 + |∇u|2 + V (x)u2)dx <∞
}

be equipped with the inner product and norm

〈u, v〉 =

∫
R3

(∆u∆v +∇u∇v + V (x)uv)dx, ‖u‖ = 〈u, u〉1/2.

For λ > 0, we also need the following inner product and norm

〈u, v〉λ =

∫
R3

(∆u∆v +∇u∇v + λV (x)uv)dx, ‖u‖λ = 〈u, u〉1/2λ .



EJDE-2020/06 STATIONARY QUANTUM ZAKHAROV SYSTEMS 5

It is clear that ‖u‖ ≤ ‖u‖λ for λ ≥ 1. Now we set Xλ = (X, ‖u‖λ).
Applying conditions (A1) and (A2), by the Hölder, Young and Gagliardo-Nirenberg

inequalities, there exists a sharp constant Ā > 0 such that∫
R3

u2dx ≤ 1

b

∫
{V≥b}

V (x)u2dx+ (|{V < b}|
∫
R3

u4dx)1/2

≤ 1

b

∫
R3

V (x)u2dx+ Ā2|{V < b}|1/2(

∫
R3

|∆u|2dx)3/8(

∫
R3

u2dx)5/8

≤ 1

b

∫
R3

V (x)u2dx+
3Ā16/3|{V < b}|4/3

8

∫
R3

|∆u|2dx+
5

8

∫
R3

u2dx,

which shows that∫
R3

u2dx ≤ 8

3b

∫
R3

V (x)u2dx+ Ā16/3|{V < b}|4/3
∫
R3

|∆u|2dx.

Applying the above inequality leads to

‖u‖2H2

≤ (1 + Ā16/3|{V < b}|4/3)

∫
R3

|∆u|2dx+

∫
R3

|∇u|2dx+
8

3b

∫
R3

V (x)u2dx

≤ max
{

1 + Ā16/3|{V < b
}
|4/3, 8

3b
}‖u‖2.

(2.1)

This implies that the imbedding X ↪→ H2(R3) is continuous. Similar to the in-
equality (2.1), we also obtain

‖u‖2H2 ≤ (1 + Ā16/3|{V < b}|4/3)‖u‖2λ (2.2)

for

λ ≥ λ∗ :=
8

3b
(1 + Ā16/3|{V < b}|4/3)−1.

Since the imbedding H2(R3) ↪→ L∞(R3) is continuous, by(2.2), for any r ∈ [2,+∞)
one has ∫

R3

|u|rdx ≤ S−(r−2)
∞ ‖u‖rH2

≤ S−(r−2)
∞ (1 + Ā16/3|{V < b}|4/3)r/2‖u‖rλ

(2.3)

for λ ≥ λ∗.
We define the operator Φ : Xλ → H1(R3) as

Φ[u] = φK,u.

In the following lemma we state some properties of Φ without any proof. We refer
the reader to [6] for more details. These properties are useful to our study of the
problem.

Lemma 2.1. For any u ∈ Xλ, we have the following statements:

(i) Φ : Xλ → H1(R3) is continuous;
(ii) Φ maps bounded sets in Xλ into bounded sets in H1(R3);

(iii) Φ[tu] = t2Φ[u] for all t ∈ R;
(iv) Φ[u] > 0 when u 6= 0.



6 S. YAO, J. SUN, T.-F. WU EJDE-2020/06

Using the arguments in [17], by (2.3), when K ∈ L∞(R3), we have∫
R3

K(x)φK,uu
2dx ≤ ‖K‖2∞

∫
R3

|u|4dx

≤ ‖K‖2∞S−2
∞ (1 + Ā16/3|{V < b}|4/3)2‖u‖4λ,

(2.4)

and when K ∈ L2p/(p−4)(R3), we obtain∫
R3

K(x)φK,uu
2dx

≤
(∫

R3

|K|2p/(p−4)dx
)(p−4)/p(∫

R3

|u|pdx
)4/p

≤ ‖K‖2L2p/(p−4)S
−4(p−2)/p
∞

(
1 + Ā16/3|{V < b}|4/3

)2

‖u‖4λ.

(2.5)

Set

Θ =

{
‖K‖2∞S−2

∞
(
1 + Ā16/3|{V < b}|4/3

)2
for K ∈ L∞(R3),

‖K‖2
L2p/(p−4)S

−4(p−2)/p
∞

(
1 + Ā16/3|{V < b}|4/3

)2
for K ∈ L2p/(p−4)(R3).

Then it follows that ∫
R3

K(x)φK,uu
2dx ≤ Θ‖u‖4λ for λ ≥ λ∗. (2.6)

Define the Nehari manifold

Nλ,µ = {u ∈ Xλ\{0} : 〈I ′λ,µ(u), u〉 = 0}.

Thus, u ∈ Nλ,µ if and only if

‖u‖2λ −
∫
R3

K(x)φK,uu
2dx+ µ

∫
R3

|u|pdx = 0. (2.7)

By this equality and (2.6) one has

‖u‖2λ ≤ ‖u‖2λ + µ

∫
R3

|u|pdx =

∫
R3

K(x)φK,uu
2dx

≤ Θ‖u‖4λ for all u ∈ Nλ,µ.

So it leads to ∫
R3

K(x)φK,uu
2dx ≥ ‖u‖2λ ≥

1

Θ
for all u ∈ Nλ,µ. (2.8)

The Nehari manifold Nλ,µ is closely linked to the behavior of the function of the
form hu : t→ Iλ,µ(tu) as

hu(t) =
t2

2
‖u‖2λ −

t4

4

∫
R3

K(x)φK,uu
2dx+

µtp

p

∫
R3

|u|pdx for t > 0.

For u ∈ X, we find that

h′u(t) = t‖u‖2λ − t3
∫
R3

K(x)φK,uu
2dx+ µtp−1

∫
R3

|u|pdx,

h′′u(t) = ‖u‖2λ − 3t2
∫
R3

K(x)φK,uu
2dx+ µ(p− 1)tp−2

∫
R3

|u|pdx.

This implies that for u ∈ X\{0} and t > 0, h′u(t) = 0 holds if and only if tu ∈ Nλ,µ

by Lemma 2.1. In particular, h′u(1) = 0 holds if and only if u ∈ Nλ,µ. So, Nλ,µ



EJDE-2020/06 STATIONARY QUANTUM ZAKHAROV SYSTEMS 7

can be split into three parts corresponding to the local minima, local maxima and
points of inflection. According to [18], we define

N+
λ,µ = {u ∈ Nλ,µ : h′′u(1) > 0},

N0
λ,µ = {u ∈ Nλ,µ : h′′u(1) = 0},

N−λ,µ = {u ∈ Nλ,µ : h′′u(1) < 0}.

Then using the argument in Brown-Zhang [2, Theorem 2.3], we obtain the following
result.

Lemma 2.2. Suppose that u0 is a local minimizer for Iλ,µ on Nλ,µ and that u0 /∈
N0
λ,µ. Then I ′λ,µ(u0) = 0 in X−1.

For each u ∈ Nλ,µ it holds

h′′u(1) = ‖u‖2λ − 3

∫
R3

K(x)φK,uu
2dx+ µ(p− 1)

∫
R3

|u|pdx

= −2‖u‖2λ + µ(p− 4)

∫
R3

|u|pdx

= (2− p)‖u‖2λ + (p− 4)

∫
R3

K(x)φK,uu
2dx.

(2.9)

Then we have the following result.

Lemma 2.3. Suppose that p > 4,K ∈ L∞(R3)∪L2p/(p−4)(R3) with K(x) ≥ 0 and
conditions (A1) and (A2) hold. Then Iλ,µ is coercive and bounded below on N−λ,µ
for all λ ≥ λ∗ and µ > 0.

Proof. By (2.7), (2.8) and (2.9) one has

Iλ,µ(u) =
p− 2

2p
‖u‖2λ −

p− 4

4p

∫
R3

K(x)φK,uu
2dx ≥ p− 2

4p
‖u‖2λ ≥

p− 2

4pΘ
,

which implies that Iλ,µ is coercive and bounded below on N−λ,µ for all λ ≥ λ∗. �

Now, we consider the biharmonic equation

∆2u−∆u =
1

4π
K(x)

(∫
Ω

K(y)u2(y)

|x− y| exp(|x− y|)
dy
)
u in Ω,

u =
∂u

∂n
= 0 on ∂Ω,

(2.10)

where Ω is given in condition (A2) and K ∈ L∞(R3)∪L2p/(p−4)(R3) with K(x) ≥ 0.
It is easy to verify that (2.10) admits ground state solution with positive energy by
using the standard Nehari manifold method. Let ω be the ground state solution of
(2.10) and

α = inf
u∈M

J(u) = J(ω) > 0,

where J is the energy functional related with (2.10) in H2
0 (Ω) given by

J(u) =
1

2

∫
Ω

(|∆u|2 + |∇u|2)dx− 1

4

∫
Ω

K(x)φK,ωω
2dx

and M = {u ∈ H2
0 (Ω)\{0} : 〈J ′(u), u〉 = 0}. Then it holds

α =
1

2

∫
Ω

(|∆ω|2 + |∇ω|2)dx− 1

4

∫
Ω

K(x)φK,ωω
2dx



8 S. YAO, J. SUN, T.-F. WU EJDE-2020/06

=
1

4

∫
Ω

(|∆ω|2 + |∇ω|2)dx.

For any u ∈ Nλ,µ with Iλ,µ(u) < (p−2)2

p(p−4)α, we have

(p− 2)2

p(p− 4)
α >

1

2
‖u‖2λ −

1

4

∫
R3

K(x)φK,uu
2dx+

µ

p

∫
R3

|u|pdx

=
1

4
‖u‖2λ −

µ(p− 4)

4p

∫
R3

|u|pdx

≥ 1

4
‖u‖2λ −

µ(p− 4)

4pSp−2
∞

(
1 + Ā16/3|{V < b}|4/3

)p/2
‖u‖pλ

for λ ≥ λ∗. This indicates that for each λ ≥ λ∗ and 0 < µ < 2(p−2)/2µ∗, there exist
two constants D,D > 0 satisfying

2
[α(p− 2)2

p(p− 4)

]1/2
< D < 2

[α(p− 2)

p− 4

]1/2
< D (2.11)

such that

‖u‖λ < D or ‖u‖λ > D.

Hence, we obtain

Nλ,µ

( (p− 2)2

p(p− 4)
α
)

:=
{
u ∈ Nλ,µ : Iλ,µ(u) <

(p− 2)2

p(p− 4)
α
}

=N
(1)
λ,µ ∪N

(2)
λ,µ,

where

N
(1)
λ,µ =

{
u ∈ Nλ,µ

( (p− 2)2

p(p− 4)
α
)

: ‖u‖λ < D}

and

N
(2)
λ,µ =

{
u ∈ Nλ,µ

( (p− 2)2

p(p− 4)
α
)

: ‖u‖λ > D
}
.

This shows that

‖u‖λ < D < 2
[α(p− 2)

p− 4

]1/2
for all u ∈ N

(1)
λ,µ,

‖u‖λ > D > 2
[α(p− 2)

p− 4

]1/2
for all u ∈ N

(2)
λ,µ.

It follows from (2.9) and (2.11) that

h′′u(1) = −2‖u‖2λ + µ(p− 4)

∫
R3

|u|pdx

≤ −2‖u‖2λ + µ(p− 4)S−(p−2)
∞ (1 + Ā16/3|{V < b}|4/3)p/2‖u‖pλ

< −2‖u‖2λ + 2
[ (p− 4)

4(p− 2)α

](p−2)/2‖u‖pλ < 0 for u ∈ N
(1)
λ,µ.

Moreover,

p− 2

2p
‖u‖2λ −

p− 4

4p

∫
R3

K(x)φK,uu
2dx = Iλ,µ(u)

<
(p− 2)2

p(p− 4)
α



EJDE-2020/06 STATIONARY QUANTUM ZAKHAROV SYSTEMS 9

<
p− 2

4p
‖u‖2λ for u ∈ N

(2)
λ,µ,

and so

h′′u(1) = (2− p)‖u‖2λ − (4− p)
∫
R3

K(x)φK,uu
2dx > 0 for u ∈ N

(2)
λ,µ.

Hence, the following statement is true.

Lemma 2.4. If p > 4, λ ≥ λ∗ and 0 < µ < 2(p−2)/2µ∗, then N
(1)
λ,µ ⊂ N−λ,µ and

N
(2)
λ,µ ⊂ N+

λ,µ are C1 sub-manifolds. Furthermore, each local minimizer of the

functional Iλ,µ on both N
(1)
λ,µ and N

(2)
λ,µ is a critical point of Iλ,µ in X.

For u ∈ Xλ\{0}, we define

T (u) =
( ‖u‖2λ∫

R3 K(x)φK,uu2dx

)1/2

.

Lemma 2.5. Suppose that p > 4, K ∈ L∞(R3)∪L2p/(p−4)(R3) with K(x) ≥ 0 and
conditions (A1) and (A2) hold. Then for each µ > 0 and u ∈ Xλ\{0} satisfying∫

R3

K(x)φK,uu
2dx

>
2(p− 2)

p− 4
(
µ(p− 4)

2Sp−2
∞

)2/(p−2)(1 + Ā16/3|{V < b}|4/3)p/(p−2)‖u‖4λ,

there exists a constant t̂(2) > ( 2(p−2)
p−4 )1/2T (u) such that

inf
t≥0

Iλ,µ(tu) = inf
(
2(p−2)
p−4 )1/2T (u)<t<t̂(2)

Iλ,µ(tu) < 0.

Proof. For any u ∈ Xλ\{0} and t > 0, we have

Iλ,µ(tu) = tp
[ t2−p

2
‖u‖2λ −

t4−p

4

∫
R3

K(x)φK,uu
2dx+

µ

p

∫
R3

|u|pdx
]
.

Let

l(t) =
t2−p

2
‖u‖2λ −

t4−p

4

∫
R3

K(x)φK,uu
2dx.

Clearly, Iλ,µ(tu) = 0 if and only if

l(t) +
µ

p

∫
R3

|u|pdx = 0.

It is easily seen that

l(t0) = 0, lim
t→0+

l(t) =∞ and lim
t→∞

l(t) = 0,

where t0 =
√

2T (u). Considering the derivative of l(t), we obtain

l′(u) = − (q − 2)t1−q

2
‖u‖2λ +

(p− 4)t2p−q−1

4

∫
R3

K(x)φK,uu
2dx

= t1−q
[ (p− 4)t2

4

∫
R3

K(x)φK,uu
2dx− (q − 2)

2
‖u‖2λ

]
.



10 S. YAO, J. SUN, T.-F. WU EJDE-2020/06

This indicates that l(t) is decreasing when 0 < t < ( 2(p−2)
p−4 )1/2T (u) and is increasing

when t >
( 2(p−2)

p−4

)1/2
T (u), and hence

inf
t>0

l(t) = − 1

p− 4

[ 2(p− 2)‖u‖2λ
(p− 4)

∫
R3 K(x)φK,uu2dx

]−(p−2)/2

‖u‖2λ.

For each u ∈ Xλ\{0} satisfying∫
R3

K(x)φK,uu
2dx

>
2(p− 2)

p− 4

(µ(p− 4)

2Sp−2
∞

)2/(p−2)(
1 + Ā16/3|{V < b}|4/3

)p/(p−2)

‖u‖4λ,

by (2.3) one has

inf
t>0

l(t) = − 1

p− 4

[ 2(p− 2)‖u‖2λ
(p− 4)

∫
R3 K(x)φK,uu2dx

]−(p−2)/2

‖u‖2λ

< − µ

pSp−2
∞

(1 + Ā16/3|{V < b}|4/3)p/2‖u‖pλ

< −µ
p

∫
R3

|u|pdx,

which implies that there exist two numbers t̂(i) (i = 1, 2) satisfying

0 < t̂(1) <
(2(p− 2)

p− 4

)1/2

T (u) < t̂(2)

such that

Iλ,µ(t̂(i)u) = 0 for i = 1, 2.

Moreover,

Iλ,µ
[(2(p− 2)

p− 4

)1/2
T (u)u

]
< 0,

and so inft≥0 Iλ,µ(tu) < 0. Note that

h′u(t) = ptp−1
[
l(t) +

µ

p

∫
R3

|u|pdx
]

+ tpl′(t),

leading to

h′u(t) < 0 for all t ∈
[
t̂(1), big(

2(p− 2)

p− 4

)1/2
T (u)

]
and h′u(t̂(2)) > 0.

The proof is complete. �

Lemma 2.6. Suppose that p > 4, K ∈ L∞(R3)∪L2p/(p−4)(R3) with K(x) ≥ 0 and
conditions (A1) and (A2) hold. Then for each µ > 0 and u ∈ Xλ\{0} satisfying∫

R3

K(x)φK,uu
2dx

>
2(p− 2)

p− 4

(µ(p− 4)

2Sp−2
∞

)2/(p−2)(
1 + Ā16/3|{V < b}|4/3

)p/(p−2)

‖u‖4λ,

there are two positive constants t+(u) and t−(u) satisfying

T (u) < t−(u) <
(p− 2

p− 4

)1/(2p−2)
T (u) < t+(u)



EJDE-2020/06 STATIONARY QUANTUM ZAKHAROV SYSTEMS 11

such that t±(u)u ∈ N±λ,µ and Iλ,µ(t−(u)u) = sup0≤t≤t+(u) Iλ,µ(tu) and

Iλ,µ(t+(u)u) = inf
t≥t−(u)

Iλ,µ(tu) = inf
t≥0

Iλ,µ(tu) < 0.

Proof. Define

g(t) = t2−p‖u‖2λ − t4−p
∫
R3

K(x)φK,uu
2dx for t > 0.

Clearly, tu ∈ Nλ,µ if and only if g(t)+µ
∫
R3 |u|pdx = 0. A straightforward evaluation

shows that
g(T (u)) = 0, lim

t→0+
g(t) =∞, lim

t→∞
g(t) = 0.

Note that

g′(t) = t1−p
[
− (p− 2)‖u‖2λ + (p− 4)t2

∫
R3

K(x)φK,uu
2dx
]
.

Then we obtain that g(t) is decreasing when 0 < t < (p−2
p−4 )1/2T (u) and is increasing

when t > (p−2
p−4 )1/2T (u), which implies that

inf
t>0

g(t) = g
((p− 2

p− 4

)1/2
T (u)

)
.

For each u ∈ Xλ\{0} satisfying∫
R3

K(x)φK,uu
2dx

>
2(p− 2)

p− 4

(µ(p− 4)

2Sp−2
∞

)2/(p−2)

(1 + Ā16/3|{V < b}|4/3)p/(p−2)‖u‖4λ,

it follows from (2.3) that

g
(

(
p− 2

p− 4
)1/2T (u)

)
= − 2

p− 4

[ (p− 2)‖u‖2λ
(p− 4)

∫
R3 K(x)φK,uu2dx

](2−p)/2‖u‖2λ
< −µS−(p−2)

∞
(
1 + Ā16/3|{V < b}|4/3

)p/2‖u‖pλ
≤ −µ

∫
R3

|u|pdx.

Then there exist two constants t+(u) and t−(u) such that

T (u) < t−(u) < (
p− 2

p− 4
)1/2T (u) < t+(u),

g(t±(u)) + µ

∫
R3

|u|pdx = 0.

Namely, t±(u)u ∈ Nλ,µ. By a calculation on the second order derivatives, we find
that

h′′t−(u)u(1) = (t−(u))p+1g′(t−(u)) < 0,

h′′t+(u)u(1) = (t+(u))p+1g′(t+(u)) > 0.

These imply that t±(u)u ∈ N±λ,µ. It is easily seen that h′u(t) > 0 holds for all

t ∈ (0, t−(u)) ∪ (t+(u),∞) and h′u(t) < 0 holds for all t ∈ (t−(u), t+(u)), which
leads to

Iλ(t−(u)u) = sup
0≤t≤t+(u)

Iλ(tu) and Iλ,µ(t+(u)u) = inf
t≥t−(u)

Iλ,µ(tu),



12 S. YAO, J. SUN, T.-F. WU EJDE-2020/06

and so Iλ,µ(t+(u)u) < Iλ,µ(t−(u)u). By Lemma 2.5, we have

Iλ,µ(t+(u)u) = inf
t≥t−(u)

Iλ(tu) = inf
t≥0

Iλ(tu) < 0.

This completes the proof. �

Since ω is the ground state solution of (2.10) with J(ω) = α > 0, for 0 < µ < µ∗
we have∫

R3

K(x)φK,ωω
2dx

= ‖ω‖2λ = 4α

>
2(p− 2)

p− 4

(µ(p− 4)

2Sp−2
∞

)2/(p−2)(
1 + Ā16/3|{V < b}|4/3

)p/(p−2)

‖ω‖4λ.

Then by Lemma 2.6, there exist two positive numbers t−(ω) and t+(ω) such that

1 < t−(ω) < (
p− 2

p− 4
)1/2 < t+(ω)

and t±(ω)ω ∈ N±λ,µ. Furthermore, we have

Iλ,µ(t−(ω)ω) = sup
0≤t≤t+(ω)

Iλ,µ(tω),

Iλ,µ(t+(ω)ω) = inf
t≥t−(ω)

Iλ,µ(tω) = inf
t≥0

Iλ,µ(tω) < 0,

which implies that t+(ω)ω ∈ N
(2)
λ,µ. A direct calculation shows that

Iλ,µ(t−(ω)ω) =
p− 2

2p
‖t−(ω)ω‖2λ −

p− 4

4p

∫
R3

K(x)φK,t−(ω)ω(t−(ω)ω)2dx

=
(t−(ω))2

4p

[
2(p− 2)− (p− 4)(t−(ω))2

]
‖ω‖2λ

<
(p− 2)2

p(p− 4)
α.

This indicates that t−(ω)ω ∈ N
(1)
λ,µ.

We define
γ−λ,µ = inf

u∈N(1)
λ,µ

Iλ,µ(u) = inf
u∈N−λ,µ

Iλ,µ(u).

It follows from Lemma 2.3 and the property of ω that

p− 2

4pΘ
< γ−λ,µ <

(p− 2)2

p(p− 4)
α.

We define Ψ : Xλ → R by

Ψ(u) =

∫
R3

K(x)φK,uu
2dx.

We now show that the functional Ψ and its derivative Ψ′ have Brezis-Lieb splitting
property.

Lemma 2.7. Assume that K ∈ L∞(R3) ∪ L2p/(p−4)(R3) with K(x) ≥ 0. Let
un ⇀ u in Xλ and un → u a.e. in R3. Then as n → ∞, the following statements
hold:

(i) Ψ(un − u) = Ψ(un)−Ψ(u) + o(1);



EJDE-2020/06 STATIONARY QUANTUM ZAKHAROV SYSTEMS 13

(ii) Ψ′(un − u) = Ψ′(un)−Ψ′(u) + o(1) in X−1
λ .

The proof of the above lemma is similar to that of [19, Lemma 4.2], we omit it
here.

3. Proof of Theorem 1.1

First we investigate the compactness condition for the functional Iλ,µ.

Proposition 3.1. Suppose that p > 4, K ∈ L∞(R3)∪L2p/(p−4)(R3) with K(x) ≥ 0
and conditions (A1) and (A2) hold. Then there exists Λ∗ > λ∗ such that if {un} ⊂
N

(1)
λ,µ is a (PS)β-sequence for Iλ,µ with β < (p−2)2

p(p−4)α, then {un} converges strongly

in X up to subsequence for all λ > Λ∗.

Proof. Let {un} ⊂ N
(1)
λ,µ be a (PS)β-sequence for Iλ,µ with β < (p−2)2

p(p−4)α. It is clear

that {un} is bounded in Xλ. Then there exist a subsequence {un} and u0 in Xλ

such that

un ⇀ u0 weakly in Xλ;

un → u0 strongly in Lrloc(R3) for 2 ≤ r <∞;

un(x)→ u0(x) a.e. on R3.

Moreover, I ′λ,µ(u0) = 0 and ‖u0‖λ ≤ lim infn→∞ ‖un‖λ < D. Let vn = un − u0.
Then vn ⇀ 0 in Xλ and

‖vn‖λ ≤ 2D + o(1). (3.1)

It follows from condition (A1) that∫
R3

v2
ndx ≤

1

λb

∫
R3

λV (x)v2
ndx+

∫
{V <b}

v2
ndx ≤

1

λb
‖vn‖2λ + o(1).

From this inequality,(2.2) and the Sobolev inequality, for r > 2 we have∫
R3

|vn|rdx ≤ |vn|r−2
∞

∫
R3

v2
ndx

≤ S−(r−2)
∞ ‖vn‖r−2

H2 ·
∫
R3

v2
ndx

≤ 1

λb
S−(r−2)
∞ (1 + Ā16/3|{V < b}|4/3)(r−2)/2‖vn‖rλ + o(1).

(3.2)

When K ∈ L∞(R3), from (2.4) and (3.2) it follows that∫
R3

K(x)φK,vnv
2
ndx ≤ ‖K‖2∞

∫
R3

|vn|4dx

≤ 1

λb
‖K‖2∞S−2

∞

(
1 + Ā16/3|{V < b}|4/3

)
‖vn‖4λ + o(1).

When K ∈ L2p/(p−4)(R3), by (2.5) and (3.2) one has∫
R3

K(x)φK,vnv
2
ndx

≤ ‖K‖2L2p/(p−4)

(∫
R3

|vn|pdx
)4/p

≤ (
1

λb
)4/p‖K‖2L2p/(p−4)S

−4(p−2)/p
∞

(
1 + Ā16/3|{V < b}|4/3

)2(p−2)/p‖vn‖4λ + o(1).



14 S. YAO, J. SUN, T.-F. WU EJDE-2020/06

Let

Πλ =


1
λb‖K‖

2
∞S
−2
∞ (1 + Ā16/3|{V < b}|4/3)

if K ∈ L∞(R3),(
1
λb

)4/p‖K‖2
L2p/(p−4)S

−4(p−2)/p
∞

(
1 + Ā16/3|{V < b}|4/3

)2(p−2)/p

if K ∈ L2p/(p−4)(R3).

Clearly, Πλ → 0 as λ→∞. Then∫
R3

K(x)φK,vnv
2
ndx ≤ Πλ‖vn‖4λ + o(1). (3.3)

Thus, from Lemma 2.7, (3.1) and (3.3) it follows that

o(1) = ‖vn‖2λ −
∫
R3

K(x)φK,vnv
2
ndx+ µ

∫
R3

|vn|pdx

≥ ‖vn‖2λ −Πλ‖vn‖4λ + o(1)

≥ ‖vn‖2λ(1−ΠλD
2) + o(1),

which implies that there exists Λ∗ > λ∗ such that vn → 0 strongly in Xλ for λ > Λ∗.
This completes the proof. �

Now, we are ready to prove Theorem 1.1. By Lemma 2.3 and the Ekeland

variational principle [5], there exists a minimizing sequence {un} ⊂ N
(1)
λ,µ such that

Iλ,µ(un) = γ−λ,µ + o(1) and I ′λ,µ(un) = o(1) in X.

It follows from Proposition 3.1 and 0 < γ−λ,µ <
(p−2)2

p(p−4)α that there exist a subse-

quence {un} and u−λ,µ ∈ X\{0} such that un → u−λ,µ strongly in Xλ for all λ > Λ∗

and 0 < µ < µ∗. Thus, u−λ,µ is a minimizer for Iλ,µ on N
(1)
λ,µ. This indicates that u−λ,µ

is a critical point of Iλ,µ by Lemma 2.4. Hence, (u−λ,µ, φK,u−λ,µ
) ∈ H2(R3)×H1(R3)

is a nontrivial solution of system (Zλ,µ).

4. Proof of Theorem 1.2

We define

γ+
λ,µ = inf

u∈N(2)
λ,µ

Iλ,µ(u) = inf
u∈N+

λ,µ

Iλ,µ(u).

Lemma 4.1. Suppose that p > 4,K ∈ L2p/(p−4)(R3) with K(x) ≥ 0 and condi-
tions(A1) and (A2) hold. Then for λ ≥ λ∗ and µ > 0, the following statements are
true:

(i) N+
λ,µ is a bounded set;

(ii) there exists a positive constant D0 such that 0 > γ+
λ > −D0.

Proof. (i) Let u ∈ N+
λ,µ. By condition (A1) and (2.5), we obtain

1 =

∫
R3 K(x)φK,uu

2dx

‖u‖2λ + µ
∫
R3 |u|pdx

<
(
∫
R3 |K|2p/(p−4)dx)(p−4)/p(

∫
R3 |u|pdx)4/p

µ
∫
R3 |u|pdx



EJDE-2020/06 STATIONARY QUANTUM ZAKHAROV SYSTEMS 15

=
(
∫
R3 |K|2p/(p−4)dx)(p−4)/p

µ(
∫
R3 |u|pdx)(p−4)/p

,

which implies that there exists a constant d1 > 0, depending on µ such that∫
R3

|u|pdx ≤ d1 for u ∈ N+
λ,µ. (4.1)

Thus, according to (2.9) one has

‖u‖2λ <
µ(p− 4)

2

∫
R3

|u|pdx ≤ µ(p− 4)

2
d1 for u ∈ N+

λ,µ.

This indicates that N+
λ,µ is a bounded set.

(ii) Let u ∈ N+
λ,µ. From Lemma 2.6, we have γ+

λ,µ < 0. Using (4.1) gives

Iλ,µ(u) =
1

4
‖u‖2λ −

µ(p− 4)

4p

∫
R3

|u|pdx

> −µ(p− 4)

4p

∫
R3

|u|pdx

≥ −µ(p− 4)

4p
d1,

which shows that there exists a constant D0 > 0 such that γ+
λ,µ > −D0 for all

λ ≥ λ∗. This completes the proof. �

Similar to Proposition 3.1, we can establish a compactness result for the func-

tional Jλ,a in N
(2)
λ,µ.

Proposition 4.2. Suppose that p > 4, K ∈ L2p/(p−4)(R3) with K(x) ≥ 0 and
conditions (A1) and (A2) hold. Then there exists a number Λ∗∗ ≥ λ∗ such that Iλ,µ

satisfies (PS)β-condition in N
(2)
λ,µ with β < (p−2)2

p(p−4)α for all λ ≥ Λ∗∗ and 0 < µ < µ∗.

Now, we are ready to proof Theorem 1.2. Similar to the argument of Theorem
1.1, we obtain that u−λ,µ is a critical point of Iλ,µ satisfying Iλ,µ(u−λ,µ) = γ−λ,µ =

inf
u∈N(1)

λ,µ

Iλ,µ(u) > 0 for all λ > Λ∗ and 0 < µ < µ∗.

By Lemma 4.1 and the Ekeland variational principle [5], there exists a minimizing

sequence {un} ⊂ N
(2)
λ,µ such that

Iλ,µ(un) = γ−λ,µ + o(1) and I ′λ,µ(un) = o(1) in X.

From Proposition 4.2 there exist a subsequence {un} and u+
λ,µ ∈ X\{0} such that

un → u+
λ,µ strongly in Xλ for all λ > Λ∗∗. Thus, u+

λ,µ is a minimizer for Iλ,µ on

N
(2)
λ,µ. Hence, u+

λ,µ is a critical point of Iλ,µ by Lemma 2.4. Note that

γ+
λ,µ = Iλ,µ(u+

λ,µ) ≤ Iλ,µ(t+ω) < 0,

implying u+
λ,µ ∈ N

(2)
λ . Therefore, we conclude that for λ > Λ := max{Λ∗,Λ∗∗},

system (Zλ,µ) admits at least two nontrivial solutions (u±λ,µ, φK,u±λ,µ
) ∈ H2(R3) ×

H1(R3) satisfying

0 < ‖u−λ,µ‖λ < 2
[α(p− 2)

p− 4

]1/2
< ‖u+

λ,µ‖λ



16 S. YAO, J. SUN, T.-F. WU EJDE-2020/06

and

Iλ,µ(u+
λ,µ) < 0 < Iλ,µ(u−λ,µ) <

(p− 2)2

p(p− 4)
α.

In particular, u+
λ,µ is a ground state solution.

5. Concentration of solutions

Proof of Theorem 1.3. We follow the arguments in [1, 13]. Choosing a positive
sequence {λn} such that Λ < λ1 ≤ λ2 ≤ · · · ≤ λn →∞ as n→∞. Let u±n := u±λn,µ

be the solutions obtained in Theorem 1.2 with u−λn,µ ∈ N
(1)
λn,µ

and u+
λn,µ

∈ N
(2)
λn,µ

.

By Lemma 4.1 (i) and the definition of N
(1)
λn,µ

, there exists a constant M > 0,

independent of λn such that ‖u±n ‖λn ≤ M , leading to ‖u±n ‖λ1
≤ M . Thus, there

exist u±∞ ∈ X such that

u±n ⇀ u±∞ weakly in Xλ1 ;

u±n → u±∞ strongly in Lrloc(R3) for 2 ≤ r <∞;

u±n (x)→ u±∞(x) a.e. on R3.

Similar to the proof of Proposition 3.1, we conclude that

u±n → u±∞ strongly in Xλ1
.

This shows that u±n → u±∞ strongly in H2(R3) by (2.2). By Fatou’s Lemma, we
obtain ∫

R3

V (x)(u±∞)2dx ≤ lim inf
n→∞

∫
R3

V (x)(u±n )2dx ≤ lim inf
n→∞

‖u±n ‖2λn
λn

= 0,

which implies that u±∞ = 0 a.e. in R3\Ω. Moreover, fixing ϕ ∈ C∞0 (R3\Ω), we have∫
R3\Ω

∇u±∞(x)ϕ(x)dx = −
∫
R3\Ω

u±∞(x)∇ϕ(x)dx = 0,

which indicates that ∇u±∞(x) = 0 a.e. in R3\Ω. Since ∂Ω is smooth, u±∞ ∈
H2(R3\Ω) and ∇u±∞ ∈ H1(R3\Ω), it follows from Trace Theorem that there are

constants C, C̃ > 0 such that

‖u±∞‖L2(∂Ω) ≤ C‖u±∞‖H2(R3\Ω) = 0,

‖∇u±∞‖L2(∂Ω) ≤ C̃‖∇u±∞‖H1(R3\Ω) = 0.

These show that u±∞ ∈ H2
0 (Ω).

Since 〈Iλn,µ(u±n ), ϕ〉 = 0 for any ϕ ∈ C∞0 (Ω), it is easy to verify that∫
Ω

(∆u±∞∆ϕ+∇u±∞∇ϕ)dx+ µ

∫
Ω

|u±∞|p−2u±∞ϕdx =

∫
Ω

K(x)φK,u±∞u
±
∞ϕdx.

This tells us that u±∞ are weak solutions of (1.6) by the denseness of C∞0 (Ω) in
H2

0 (Ω). By (2.8) and the facts of u±n → u±∞ strongly in Xλ1
and u±∞ ∈ H2

0 (Ω), we
have ∫

Ω

(|∆u±∞|2 + |∇u±∞|2)dx ≥ 1

Θ
> 0.

This implies that u±∞ 6= 0. Furthermore,

Iλn,µ(u+
n ) < 0 <

p− 2

4pΘ
< Iλn,µ(u−n ) <

(p− 2)2

p(p− 4)
α.



EJDE-2020/06 STATIONARY QUANTUM ZAKHAROV SYSTEMS 17

Then u+
∞ 6= u−∞. This completes the proof. �

Acknowledgments. J. Sun was supported by the National Natural Science Foun-
dation of China (Grant No. 11671236). T. F. Wu was supported by the Ministry
of Science and Technology, Taiwan (Grant No. 108-2115-M-390-007-MY2), and by
the National Center for Theoretical Sciences, Taiwan.

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Shuai Yao

School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049,
China

Email address: shyao2019@163.com



18 S. YAO, J. SUN, T.-F. WU EJDE-2020/06

Juntao Sun

School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049,

China.
School of Mathematical Sciences, Qufu Normal University, Shandong 273165, China

Email address: jtsun@sdut.edu.cn

Tsung-Fang Wu

Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811,

Taiwan
Email address: tfwu@nuk.edu.tw


	1. Introduction
	2. Preliminaries
	3. Proof of Theorem ??
	4. Proof of Theorem ??
	5. Concentration of solutions
	Acknowledgments

	References