Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 79, pp. 1–12.

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

PERIOD FUNCTIONS AND CRITICAL PERIODS OF

PIECEWISE LINEAR SYSTEM

SHAOQING WANG, JIAZHONG YANG

Communicated by Shaobo Gan

Abstract. In this article, we study a rich and complex phenomena of planar
piecewise linear systems having two domains of which the separation curves

are not straight. We show that for positive integers n and m and non-negative

integers n1, n2, . . . , nm, there exist two types of piecewise linear systems: one
has a period annulus possessing exactly n critical periods; the other has m

period annuli possessing exactly n1, n2, . . . , nm critical periods. Moreover, an

algebraic curve can be chosen as the separation line in system of the first type.

1. Introduction

One of the important problems in qualitative theory of planar differential equa-
tions is studying centers of systems. Here an isolated singularity O is called a center
if there is a neighborhood U of O where all trajectories in U \ O are closed. By a
period annulus, we mean the largest such neighborhood. A period annulus Ω of a
center O can be parameterized by an analytic curve Γ which is transversal to the
orbits in Ω and we denote by T (ξ), called period function, the minimal positive
period of the orbit passing through ξ ∈ Γ. In the literature, value ξ0 ∈ Γ is said
to be a critical period if T ′(ξ0) = 0. It is not difficult to verify that the number of
critical periods is independent of the choice of its parametrization. In particular,
O is called an isochronous center if T (ξ) is a constant function.

Related to the center and period function, many efforts are made to determine
the uniform maximal number H(n) of critical periods, analogue to Hilbert’s six-
teenth problem which asks for the uniform upper bound of the number of limit cycles
(see [9] and reference therein), in all polynomial systems ẋ = Pn(x, y), ẏ = Qn(x, y),
where Pn(x, y), Qn(x, y) are two polynomials of degree at most n. Meanwhile, many
interesting results have been obtained. For example, the authors in [1] proved that
for any given polynomial system, there are at most a finite number of critical pe-
riods in a period annulus contained in a compact region. And the lower bound of
H(n), found in [2] to be linear with respect to n, was improved later in [6] to be a
quadratic function of n.

2010 Mathematics Subject Classification. 34C05, 34C25, 34C27.
Key words and phrases. Piecewise linear system; period annulus; period function;

critical period.
c©2020 Texas State University.

Submitted June 22, 2020. Published July 27, 2020.

1



2 S. WANG, J. YANG EJDE-2020/79

Attention is also paid to other aspects of center and period function, such as to
isochronicity which can be seen in [3, 8, 10] and reference therein, to bifurcation of
critical periods (see for instance [7]), as well as to application (see [5, 11]).

In general, more complex phenomena could appear in planar polynomial systems
as their degrees increase, meanwhile a planar piecewise linear system with a com-
plicated separation line can also present rich dynamic behaviors, see for instance
[13] about number of limit cycles in a piecewise linear system of which the sepa-
ration line is not a straight line (see its definition below). By the way, piecewise
smooth perturbations of an isochronous center also draw some attention, readers
are referred to [12].

Although there are many attractive results about center and critical periods in
planar polynomial systems, to the best of our knowledge, only a few related conclu-
sions have been drawn in planar piecewise systems, especially in planar piecewise
linear systems. Thus in this paper, the authors focus on the latter. More precisely,
this paper aims at the number of critical periods of the piecewise linear systems.

Here, for convenience, we give some notation of piecewise systems, which can be
traced back to Filippov [4]. If the whole plane R2 is divided into two domains I1
and I2 by a continuous line γ called a separation line in this paper, and smooth
systems X1 and X2 are set in I1 and I2 respectively, then a piecewise smooth system
X = (X1, X2) comes into being. In particular, X = (X1, X2) is called a piecewise
linear system provided that X1, X2 are both linear systems.

As for the points on the separation line γ, they can be classified into four types:
sliding points;escaping points;sewing points, and tangent points. By a sliding point
(resp, escaping point), we mean a point p ∈ γ such that X1(p) and X2(p) point
inward (resp, outward) γ. A point p ∈ γ is said to be a sewing point if X1(p) and
X2(p) are transversal to γ and point to the same direction. And a tangent point
p ∈ γ means either X1(p) or X2(p) is tangent to γ.

In a piecewise system, a period annulus, similar to that in an analytic system, is
also a simply connected neighbourhood of a point O, called a Σ-center, in which all
orbits are closed around O and intersect the separation line only at sewing points.
Similarly, period function and critical period can also be defined in a piecewise
system. With these notation, three main results of this paper can be stated as
follows:

Theorem 1.1. For any integer n ≥ 1, there exists a piecewise linear system with
only one period annulus which has exactly n critical periods. Moreover, an algebraic
curve can be chosen as the separation line.

For the case of more than one annuli, we have another two results:

Theorem 1.2. For any integer m ≥ 2, there is a piecewise linear system with
exactly m period annuli which have no critical period, namely, the period functions
are all monotonic.

Theorem 1.3. For any integer m ≥ 2 and any m-tuple (n1, n2, . . . , nm) of non-
negative integers, there exists a piecewise linear system with exactly m period annuli
Ω1, Ω2, . . . , Ωm, which have exactly n1, n2, . . . , nm critical periods, respectively.

This article is organized as follows. In section 2, we give the proof of Theorem 1.1
while the construction processes of the piecewise linear systems in Theorem 1.2 and
1.3 are given in section 3 and 4 respectively.



EJDE-2020/79 PERIOD FUNCTIONS AND CRITICAL PERIODS 3

2. Proof of Theorem 1.1

Let f be a function defined on an interval I ⊂ R. We denote the graph of f by
Gr(f) = {(x, f(x)) : x ∈ I}. In this section, f is an even polynomial on R,

f(x) = c

n∑
i=0

ai
2i+ 1

x2i+2, (2.1)

where the coefficients a1, a2, . . . , an are chosen such that
n∑
i=0

aiy
i = (y − 12)(y − 22) . . . (y − n2), ∀y ∈ R, (2.2)

and c > 0 is a constant number to be determined below. Then the whole plane is
parted by Gr(f) into two domains: upper domain I = {(x, y) : y > f(x), x ∈ R}
and lower domain II = {(x, y) : y < f(x), x ∈ R}. See Figure 1 for its diagram.

x

y

f(x)
I

II

Figure 1. Diagram of
the piecewise linear sys-
tem in Section 2.

x

y

f(x)
f ′(x)

− x
f(x)

I

II

Figure 2.

In domains I and II, we set linear systems of center type

X1 :

{
ẋ = −ay,
ẏ = ax,

and

X2 :

{
ẋ = −by,
ẏ = bx,

where a and b are two distinct positive numbers. Thus a piecewise linear system
X = (X1, X2) of center-center type has been obtained. In the systems X1 and X2,
the trajectories are all circular arcs centered at (0, 0), consequently, the orbits of X
are also circles (see Figure 1).

Below we show that there is exactly a period annulus in X. As

f ′(x) = c

n∑
i=0

2i+ 2

2i+ 1
aix

2i+1,

and

− x

f(x)
=

{
− x
c
∑n

i=0
ai

2i+1x
2i+2 , if f(x) 6= 0,

∞, if f(x) = 0,

represent the slopes of the graph of f(x) and the orbit of X1 (or X2) at (x, f(x))
respectively (see Figure 2 for its diagram), then c > 0 can be chosen sufficiently
small such that

|f ′(x)| < | x

f(x)
|, |x| ≤ F0,



4 S. WANG, J. YANG EJDE-2020/79

where F0 is the largest positive real root of f ′(x). Since when x > F0, f ′(x) > 0 >
−x/f(x), there is a period annulus in X containing the whole plane with period
function T (x).

The following is some information about T (x) and its derivatives.

Lemma 2.1. Let Y be a linear system having a center at (0, 0),

Y :

{
ẋ = −ay,
ẏ = ax,

where a > 0. Then the time between A(x, y) (x > 0) and B(0,
√
x2 + y2) which

are in the same orbit of Y (see Figure 3) is 1
a (π2 − arctan y

x ). Moreover, the time

between C(0,−
√
x2 + y2) and A(x, y) (x > 0) is 1

a (π2 + arctan y
x ).

O
x

y

A(x, y)

B

C

Figure 3. Diagram for Lemma 2.1

Proof. The linear system Y can be transformed, by direct computation, into the
system in polar coordinates,

ṙ = 0,

θ̇ = a,

where r =
√
x2 + y2 and θ = arctan(y/x).

Since A(x, y) and B(0,
√
x2 + y2) correspond to (

√
x2 + y2, arctan(y/x)) and

(
√
x2 + y2, π2 ) respectively in polar coordinates, thus the time from A(x, y) (x > 0)

to B(0,
√
x2 + y2) is 1

a (π2 − arctan y
x ).

Similarly, the second result of this lemma holds. �

Lemma 2.2. The sign of T ′(x) depends on (f(x)/x)′.

Proof. Based on the trajectories of X1 and X2, the closed orbits of X have to
intersect the graph of f(x). By Lemma 2.1 and the even property of f(x), we have
that

T (x) = 2
(1

a

(π
2
− arctan

f(x)

x

)
+

1

b

(π
2

+ arctan
f(x)

x

))
,

then by direct computation,

T ′(x) = (
2

b
− 2

a
)

( f(x)x )′

1 + ( f(x)x )2
.

Thus the conclusion follows. �



EJDE-2020/79 PERIOD FUNCTIONS AND CRITICAL PERIODS 5

Now we prove Theorem 1.1. The expression (2.1) implies that(f(x)

x

)′
=
(
c

n∑
i=0

ai
2i+ 1

x2i+1
)′
,

= c

n∑
i=0

aix
2i,

= c(x2 − 12)(x2 − 22) . . . (x2 − n2),

(2.3)

which shows that
( f(x)

x

)′
has n distinct positive zeros that are all simple: 1, . . . , n.

We calculate the second derivative of T (x) as follows:

T ′′(x) =
(2

b
− 2

a

)( ( f(x)x )′′

1 + ( f(x)x )2
−

2( f(x)x )( f(x)x )
′2

(1 + ( f(x)x )2)2

)
. (2.4)

Since ( f(x)x )′ = 0, while ( f(x)x )′′ 6= 0, at x = 1,2,. . . ,n, combining Lemma 2.2,
(2.3) and (2.4), we obtain that 1, . . . , n are exactly n zeros of T ′(x) which are sim-
ple. Namely, the annulus in X possesses exactly n critical periods. Consequently,
Theorem 1.1 has been proved.

3. Proof of Theorem 1.2

In this section, we shall first prove Theorem 1.2 in the case of m = 3 since all
essential ideas and methods applied in the general case can be highlighted in this
special case. Here, we use another kind of piecewise linear system of focus-center
type with a more complicated separation line, which is the red line given in Figure 4.

x

y

5

4

3

2

1

−1 1

A1

B1 B2

A4A3

B3 B4

A6

A2

A5P

Q
II

I

Figure 4. Separation line in the proof of Theorem 1.2

Specifically, in Figure 4, all lines are straight lines which are parallel to the two
axes. And all points Ai, Bi (i = 1, . . . , 5) are at integer coordinates. Meanwhile
A6A5P and A1A2Q are two rays. Now the polygonal line (the red line in Figure 4)
γ := PA5A6 ∪A6B4 ∪B3B4 ∪B3A3 ∪A3A4 ∪A4B2 ∪B1B2 ∪B1A1 ∪A1A2Q, i.e.

γ := PA5A6B4B3A3A4B2B1A1A2Q

forming a separation line separates the whole plane into two parts: domain I and
II, which can be seen in Figure 4.



6 S. WANG, J. YANG EJDE-2020/79

In domain I, we set a linear system X1 of center type:

X1 :

{
ẋ = −y,
ẏ = x.

(3.1)

Meanwhile, a linear system X2 of saddle type, of which the orbits are hyperbolas,
is put in part II:

X2 :

{
ẋ = y,

ẏ = x.
(3.2)

Then we obtain a piecewise linear system X = (X1, X2), of which some trajectories
are depicted in Figure 5 (blue trajectories are in the domain I, and black ones in
domain II).

x

y

5

4

3

2

1

−1 1

A1

B1 B2

A4A3

B3 B4

A6

A2

A5P

Q
II

I

Figure 5. Some orbits of piecewise linear system X in Section 3

In more detail, the trajectories of X = (X1, X2) in Figure 5 show that A6B4,
B3B4, B3A3, A4B2, B1B2 and B1A1 all consist of sliding or escaping points (see the
dashed lines in Figure 6), consequently, the closed orbits can only intersect three
lines: PA5A6, A3A4 and A1A2Q. By symmetry of A1A2, A3A4 and A5A6 with
respect to y-axis, there exist only three period annuli Ω1, Ω2 and Ω3 containing
A1A2, A3A4 and A5A6 respectively (for details see Figure 6). Denote by Ti(x) the
period function of period annulus Ωi for each i = 1, 2, 3.

The remaining work is to show that T ′
1(x), T ′

2(x) and T ′
3(x) have no zero in

interval (0, 1). Without loss of generality, it is suffice to consider T ′
1(x). The first

task is to give the expression of T1(x).
For convenience, let these three straight lines A1A2, A3A4 and A5A6 be graphs

of three functions f1(x) ≡ 1, f2(x) ≡ 3 and f3(x) ≡ 5, x ∈ [−1, 1], respectively.
A closed orbit in the period annulus Ω1 passing through (x, f1(x)) (0 < x < 1)

consists of two trajectories L1 and L2 (see Figure 7): L1 is located in the domain I,



EJDE-2020/79 PERIOD FUNCTIONS AND CRITICAL PERIODS 7

x

y

5

4

3

2

1

−1 1

A1

B1 B2

A4A3

B3 B4

A6

A2

A5P

Q
II

I

Ω1

Ω2

Ω3

Figure 6. Three period annuli in Section 3

from (x, f1(x)) to (−x, f1(x)) and conversely, L2 is from (−x, f1(x)) to (x, f1(x)),
in domain II.

x

y

A1 A2

(x, f1(x))

Ω1

L1

L2

Figure 7. A closed orbit in Ω1

By Lemma 2.2, the time in trajectory L1 is

t1(x) = 2
(π

2
− arctan

f1(x)

x

)
, x ∈ (0, 1). (3.3)

As for the time t2(x) in L2, we have the following lemma.

Lemma 3.1.

t2(x) = ln
f1(x) + x

f1(x)− x
, x ∈ (0, 1). (3.4)

Proof. By the orthogonal transformation

u =
y − x√

2
,

v =
x+ y√

2
,

(3.5)



8 S. WANG, J. YANG EJDE-2020/79

system X2 is changed into the following standard linear system of saddle type

u̇ = −u,
v̇ = v,

(3.6)

and time t2(x) can be obtained as follows:∫ f1(x)−x√
2

f1(x)+x√
2

dt = −
∫ f1(x)−x√

2

f1(x)+x√
2

du

u
= ln

f1(x) + x

f1(x)− x
,

Thus Lemma 3.1 holds. �

From expressions (3.3) and (3.4), we obtain

T1(x) = t1(x) + t2(x) = 2(
π

2
− arctan

f1(x)

x
) + ln

f1(x) + x

f1(x)− x
, (3.7)

thus T ′
1(x) can be obtained by direct computations,

T ′
1(x) = −2

( f1(x)x )′

1 + ( f1(x)x )2
+
f1(x)− x
f1(x) + x

( f1(x)x )′( f1(x)x − 1)− ( f1(x)x )′( f1(x)x + 1)

( f1(x)x − 1)2

= −
( 2

( f1(x)x )2 + 1
+

2

( f1(x)x )2 − 1

)(f1(x)

x

)′
,

From simplicity above conclusion can be summarized in the following statement.

Lemma 3.2. For i = 1, 2, 3 and x ∈ (0, 1), T ′
i (x) has the same sign as −

( fi(x)
x

)′
.

Since f1(x) ≡ 1, f2(x) ≡ 3 and f3(x) ≡ 5, x ∈ (−1, 1), then Theorem 1.2 in the
case of m = 3, as a straightforward corollary of Lemma 3.2, has been proved.

For the general case, the same method as above can be used to construct a
separation line, as well as a piecewise linear system which has exactly m period
annuli possessing no critical period.

4. Proof of Theorem 1.3

Without loss of generality, we assume that m = 3. For fixed triple (n1, n2, n3)
of non-negative integers, below we established a piecewise linear system having 3
period annuli Ω1, Ω2 and Ω3 with exactly n1, n2 and n3 critical periods respectively.

The method of the construction of piecewise linear system here is similar to that
in the proof of Theorem 1.2, but the separation line given below is more complicated.

Here the separation line is the red curve in Figure 8. More exactly, in this figure,
two straight lines D1D2, D3D4 and two rays C5P , C2Q are parallel to x-axis, while
four lines C1D1, D2C4, C3D3 and D4C6 are straight and parallel to y-axis. The
other three curves C1C2, C3C4, C5C6 are respectively graphs of even functions
f1(x), f2(x) and f3(x) defined on different intervals, and k1, k2 and k3 are three
positive integers. Here

fi(x) =

{
ki + xgi(x), if ni 6= 0,

ki, if ni = 0,
i = 1, 2, 3, (4.1)

where

gi(x) = ki

∫ x

0

1

t2
(1− cos (k8i t

4)

1 + t4

2

)dt, x ∈
[
−

4
√
niπ

k2i
,

4
√
niπ

k2i

]
, (4.2)

and k1, k2 and k3 are sufficiently large such that the following two properties hold:



EJDE-2020/79 PERIOD FUNCTIONS AND CRITICAL PERIODS 9

x

y

k3

k2 + 1

k2

k1 + 1

k1

f3(x)

f2(x)

f1(x)
C1

D1 D2

C4C3

D3 D4

C6

C2

C5P

Q

II

I

Figure 8. Separation line in the proof of Theorem 1.3

(i) niπ/k
4
i � 1;

(ii) 1/k2i � 1/ki for i = 1, 2, 3.

Then |xgi(x)| < 1, when x ∈ [− 4
√
niπ/k

2
i ,

4
√
niπ/k

2
i ]. And the non-self-intersecting

red curve γ := PC5∪ C̃5C6∪C6D4∪ D4D3∪D3C3∪ C̃3C4∪C4D2∪D2D1∪D1C1∪
C̃1C2 ∪ C2Q, i.e.

γ := PC5C6D4D3C3C4D2D1C1C2Q

as a separation line cuts the whole plane into two domains I and II (see Figure 8).
Similar to the piecewise linear system in section 3, we also set linear systems

X1 :

{
ẋ = −y,
ẏ = x,

and

X2 :

{
ẋ = y,

ẏ = x,

in domains I and II respectively. Up to now, a piecewise linear system X =
(X1, X2) of center-saddle type has been constructed. We shall show in two steps
that there exist three period annuli Ω1, Ω2, Ω3 in X with n1, n2 and n3 critical
periods respectively.

• Step1: We show that there are three period annuli in piecewise system X. Similar
to Figure 5 in Section 3, by the direction of trajectories of X1 and X2, possible
closed orbits of X must pass through the graph of f1(x), f2(x) or f3(x). By Taylor
expansion, we have

fi(x) = ki +
2ki
3
x4 + ◦(x5), |x| � 1,

thus

f ′i(x) =
8ki
3
x3 + ◦(x4), |x| � 1,

and
x

fi(x)
=

x

ki
+ ◦(x4), |x| � 1.



10 S. WANG, J. YANG EJDE-2020/79

Thus if |x| <
4
√
niπ

k2i
, |f ′i(x)| < x

f(x) . Meanwhile, analogue to the proof of Theorem

1.1 in section 2, |f ′i(x)| (resp, | x
fi(x)
|) represents absolute value of the slope of the

graph of fi(x) (resp, orbit of X1 or X2) at (x, fi(x)) (see Figure 9 for the example
of f1(x)). Thus for i = 1, 2, 3, there must be a period annulus Ωi containing the

graph of even function fi(x) (x ∈ [−
4
√
niπ

k2i
,

4
√
niπ

k2i
]) with Σ-center (0, ki) (see Figure

10, the blue and black trajectories are in domains I and II respectively).

x

y

f1(x)
f ′
1(x)

− x
f1(x)

Figure 9.

x

y

k3

k2

k1
C1

D1 D2

C4C3

D3 D4

C6

C2

C5P

Q

II

I

Ω1

Ω2

Ω3

Figure 10. The three annuli in the proof of Theorem 1.3

• Step2. We will show that period annulus Ωi has exactly ni critical periods,
i = 1, 2, 3.

We denote by Ti(x) (i = 1, 2, 3) the period function of period annulus Ωi. By
the same method as that in the proof of Lemma 3.2, similar to formula (3.7), we
have that

T1(x) = 2(
π

2
− arctan

f1(x)

x
) + ln

f1(x) + x

f1(x)− x
. (4.3)

Thus

T ′
1(x) = −

( 2

( f1(x)x )2 + 1
+

2

( f1(x)x )2 − 1

)(f1(x)

x

)′
, (4.4)

and

T ′′
1 (x) =−

( 2

( f1(x)x )2 + 1
+

2

( f1(x)x )2 − 1

)(f1(x)

x

)′′
+
( 4

(( f1(x)x )2 + 1)2
+

4

(( f1(x)x )2 − 1)2

)(f1(x)

x

)((f1(x)

x

)′)2
.

(4.5)



EJDE-2020/79 PERIOD FUNCTIONS AND CRITICAL PERIODS 11

From (4.4), the sign of T ′
1(x) corresponding to Ω1 depends on −

( f1(x)
x

)′
. The

expression of f1(x) = k1 + xg1(x) gives(f1(x)

x

)′
=
(k1
x

)′
+ g′1(x) = −k1 − x

2g′1(x)

x2
.

With (4.2), it follows that (f1(x)

x

)′
= −k1 cos (k81x

4)

x2(1 + x4

2 )
,

which has only n1 simple zeros in (0, 4
√
n1π/k

2
1):

4
√
π/2

k21
,

4
√

3pi/2

k21
, . . . ,

4

√
(n1 − 1

2 )π

k21
.

Consequently, by (4.4) and (4.5), T ′
1(x) has only n1 zeros which are all simple

in (0, 4
√
n1π/k

2
1). Namely, the period annulus Ω1 has exactly n1 critical periods.

Similarly, Ω2 (resp, Ω3) has exactly n2 (resp, n3) critical periods.
Combining step 1 and step 2, we have proved Theorem 1.3 in the case of m = 3.

For a general m, the same method can be applied. Thus Theorem 1.3 has been
proved.

Acknowledgements. This work was supported by the National Science Founda-
tion of China (Grants NSFC 11671016 and 11471027). The authors would like to
express our gratitude to the referees for their helpful suggestions.

References

[1] C. Chicone, F. Dumortier; Finiteness for critical periods of planar analytic vector fields,

Nonlinear Anal., 20(4) (1993), 315-335.
[2] A. Cima, A. Gasull, P. R. da Silva; On the number of crital periods for planar polynomial

systems, Nonlinear Anal., 69 (2008), 1889-1903.

[3] C. Du, Y. Liu, H. Mi; A class of ninth degree system with four isochronous centers, Comp.
Math. Appl., 56 (2008), 2609-2620.

[4] A. F. Filippov; Differential equations with discontinuous right-hand sides, Nauka, Moscow,

(1985), transl, :Kluwer, Dordrecht, (1988).
[5] E. Freire, A. Gasull, A. Guillamon; First derivative of the period function with applications,

J. Diff. Equ., 204 (2004), 139-162.
[6] A. Gasull, C. Liu, J. Yang; On the number of critical periods for the planar polinomial systems

of arbitrary degree, J. Diff. Equ., 249 (2010), 684-692.

[7] A. Gasull, Y. Zhao; Bifurcation of critical periods from the rigid quadratic isochronous vector
field, Bull. Sci. math., 132 (2008), 292-312.

[8] X. Jarque, J. Villadelprat; Nonexistence of isochronous centers in planar polynomial Hamil-
tonian system of degree four, J. Diff. Equ., 180 (2002), 334-373.

[9] C. Li; Abelian integrals and limit cycles, Qual. Theory Dyn. Syst., 11 (2012), 111-128.
[10] J. Llibre, V. G. Romanovski; Isochronicity and linearizability of planar polynomial Hamil-

tonian systems, J. Diff. Equ., 259 (2015), 1649-1662.
[11] F. Rothe; The periods of the Volterra-Lotka system, J. Reine Angew. Math., 355 (1985),

129-138.
[12] H. Song, L. Peng, Y. Cui; Limit cycles in piecewise smooth perturbations of a quartic

isochronous center, Electron. J. Differential Equations, 2019, no. 107 (2019), 1-23.
[13] C. Zou, J. Yang; Piecewise linear system with a center-saddle type singularity, J. Math. Anal.

Appl., 459 (2018), 453–463.

Shaoqing Wang

School of Mathematical Sciences, Peking University, Beijing 100871, China
Email address: wangshaoqing@pku.edu.cn



12 S. WANG, J. YANG EJDE-2020/79

Jiazhong Yang

School of Mathematical Sciences, Peking University, Beijing 100871, China

Email address: jyang@math.pku.edu.cn


	1. Introduction
	2. Proof of Theorem 1.1
	3. Proof of Theorem 1.2
	4. Proof of Theorem 1.3
	Acknowledgements

	References