### 4.1. Complete Characterizations of MP and AMP Using Green Functions

Using Green functions
, we have the following characterizations on MP and AMP.

Theorem 4.1.

Let
with the Green function
. Then
admits MP iff
and

Proof.

We give only the proof for AMP.

The sufficiency is as follows. Suppose that
satisfies
. Then, for any
, it is easy to see from (3.1) that
for all
. We will show that
for all
and consequently (1.2) admits AMP.

Otherwise, suppose that

for some

, that is,

Since

, we have necessarily

From (3.24), we know that

(i)on the interval

,

is a solution of (2.1);

(ii)on the interval

,

is also a solution of (2.1).

We assert that these solutions are nonzero when the corresponding intervals are nontrivial. As

is composed of two linearly independent solutions

, the nontriviality of these solutions is the same as

which are evident because
and (3.16) shows that
.

From the above assertion, we know that
(=
has only isolated zeros for
. As
, we have
, a contradiction with (4.2).

For the necessity, let us assume that

. Then one has some

so that

. Hence one has some

such that

Let us choose

such that

Then

. However, the corresponding periodic solution

of (1.2) satisfies

Hence
does not admit AMP.

In order to apply Theorem 4.1, it is important to compute the signs of the following nonlinear functionals of potentials:

To this end, let us establish some relation between
and
.

For general

, denote

Suppose that

so that

is meaningful. We assert that

In fact, for

, the first case of (4.11) follows immediately from the defining equalities (3.24), (3.25), and (4.10). On the other hand, for

, from the second case of (3.24), one has

Hence (4.11) is also true for this case.

By introducing the domain

and the following nonlinear functionals

we have the following statements.

Lemma 4.2.

There hold, for all

,

Proof.

We only prove the first equality of (4.15) because the second one is similar. By (4.11), for any

, we have

This is just (4.15) because
.

Remark 4.3.

- (i)

- (ii)

- (iii)

By Lemma 4.2, Theorem 4.1 can be restated as follows.

Corollary 4.4.

Let
. Then
admits MP iff
, and
admits AMP iff
.

### 4.2. Complete Characterizations of MP and AMP Using Eigenvalues

Lemma 4.5.

Let
be such that
. Then
and
admits MP.

Proof.

For any
, one has
and
. See [10]. Thus
. In the following let us fix any
.

Step 1.

Since
, we can use the representation (3.35) for
where
and
are nonzero solutions of (2.1). Since
, both
have at most one zero. See Lemma 2.3. Hence
has at most two zeros. However, as
is
-periodic,
does not have any zero. This proves (4.20).

Step 2.

If (4.21) is false, there exists

such that

. By introducing

We know from (3.28) and (4.20) that

satisfies

This shows that

. Since

is a nonzero solution of (2.1), (4.23) implies

Since
has the same nonzero value at the end-points of the interval
, it is easy to see from (4.24) and (4.25) that
must have another zero
which is different from
. Consequently, the solution
of (2.1) has at least zeros
and
. This is impossible because
. See Lemma 2.3.

Step 3.

To prove (4.27), let us fix

and consider

, where

. Then

. Since

,

for all

. When

,

can be estimated. The basic idea is to consider (1.3) as a perturbation of the equation

It is well known that the difference

can be controlled by the norm of the potential

when

. For piecewise continuous and

potentials, see [

10] and [

12], respectively. Similar estimates are also true for

potentials. In fact, these can be generalized to Hill's equations with coefficients being measures [

16]. We quote from [

12, Theorem

] the following result:

as

. We conclude

On the other hand, by (4.21) and (4.26),

Moreover, it follows from Lemma 2.4 that
is continuous in
. Thus (4.27) follows simply from (4.32) and (4.33).

Step 4.

Since
,
. It follows from (4.21), (4.26), and (4.27) that, for all
,
has the same sign with
. Thus
. By Corollary 4.4,
admits MP.

Lemma 4.6.

Suppose that
satisfies
and
. Then
and
admits AMP.

Proof.

Recall from [

11] that eigenvalues

and

can be characterized using rotation numbers by

Here

is arbitrary. Hence

In the following, let

. We have

,

and

. Now we argue as in the proof of Lemma 4.5. In this case, result (4.20) can be obtained from Lemma 3.5(i) because

. If (4.21) is false at some

, we have also

. By letting

be as in (4.22), one has also some

such that

and

. With loss of generality, let us assume that

. Notice that the solution

of (2.1) has zeros

and

. By the Sturm comparison theorem, any nonzero solution

of (2.1) has at least one zero in

. In particular, for any

,

is a solution of (2.1). Hence there exists some

such that

From these, the distribution of zeros of the specific solution

satisfies

By definition (2.16) for the rotation number, we obtain

a contradiction with the characterization of
. Thus (4.21) is also true for
.

Since
, we have from (4.21) and (4.26) that
, because we will prove in Lemma 4.7 that
for all
.

Note that

is the set of potentials which are in the first ellipticity zone. By Lemmas 2.1 or 3.5,

for all

. It seems that there are several ways to deduce that

for all

. However, some remarkable result on elliptic Hill's equations by Ortega [

14,

15] can simplify the argument. Let us describe the result. Suppose that

. Consider the temporal-spatial transformation

where

and

. Then (2.1) is transformed into a new Hill's equation

where

is now

periodic. The result of Ortega shows that it is always possible to choose some

such that the Poincaré matrix

(of the period

) of (4.43) is a rigid rotation

See [15, Lemma
] and [21]. We remark that such a result is very important to study the twist character and Lyapunov stability of periodic solutions of nonlinear Newtonian equations and planar Hamiltonian systems. See, for example, [14, 15, 21].

Note that the transformation (4.42) does not change rotation numbers. Recall that the polar coordinates to define rotation numbers are

We see from (4.44) that

is related with

via

Lemma 4.7.

Proof.

We first prove that

,

, is invariant under transformations (4.42). In fact, it is well known that

and

are conjugate

for some

. Denote

From (4.49), one has the explicit relation

Note that the quadratic form

is definite. See the proof of Lemma 3.5(i). Since

, we have

Hence
is invariant under transformations (4.42).

Now (4.48) can be obtained as follows. Let
. Then
. By (4.47), the transformed potential
satisfies
. By the invariance, we have the desired result (4.48).

Lemma 4.8.

Suppose that
satisfies
. Then
and
admits AMP.

Proof.

Since

, we have

and

. See (2.11). Moreover, by (2.10), we have

. Let

. Then

for all

. We know from Lemma 4.6 that

for

. Letting

and noticing that

is continuous at

, we get

On the other hand, let us take an antiperiodic eigen function

of (2.1) associated with

. Denote by

the smallest nonnegative zero of

. Then

. Moreover, both

and

are zeros of

because of the

-antiperiodicity of

. By the Sturm comparison theorem, the solution

of (2.1) must have some zero in

. Hence

. As

, we obtain

In conclusion we have
.

Lemma 4.9.

Suppose that
satisfies
. Then
does not admit neither MP nor AMP.

Proof.

We need not to consider the case
because
is not invertible.

In the following let us assume that
satisfies
. Then
. The following is a modification of the last part of the proof of Lemma 4.8.

Let us take an antiperiodic eigenfunction

associated with

. Then the set of all zeros of

is

for some

. Denote

Then

is a nonzero solution of (2.1). Since

, by applying the Sturm comparison theorem to

and

, we know that

must have some zero

in

, the interior of the interval

because

and

are consecutive zeros of

. As

, one must have

Thus

changes sign near

. Consequently,

Now Corollary 4.4 shows that

does not admit AMP. We have also

Hence
does not admit MP.

Due to the ordering (2.10) of eigenvalues, the statements in Theorems 1.1 and 1.2 are equivalent. Now let us give the proof of Theorem 1.2. Recall that
and
for all
. By Lemma 4.5, if
,
admits MP. By Lemmas 4.6 and 4.8,
admits AMP for
. By Lemma 4.9,
does not admit MP nor AMP for
. Using the ordering (2.10) for eigenvalues, we complete the proof of Theorem 1.2.

From Lemmas 4.5, 4.6, 4.8, and 4.9, the sign of Green functions is clear in all cases.

Definition 4.10.

Given

, we say that

admits

*strong antimaximum principle* (SAMP) if

admits AMP and, moreover, there exists

such that

Then we have the following complete characterizations for SAMP.

Theorem 4.11.

Let
. Then
admits SAMP iff
iff
.

### 4.3. Explicit Conditions for AMP

Let us recall some known sufficient conditions for AMP.

Lemma 4.12 (Torres and Zhang [9]).

Suppose that

satisfies the following two conditions:

Then
admits AMP.

In the proof there, the positiveness condition (4.60) is technically used extensively. Some optimal estimates on condition (4.61) can be found in Zhang and Li [

22]. For an exponent

, let us introduce the following Sobolev constant:

Here

. These constants

can be explicitly expressed using the Gamma function of Euler. The following lower bound for

is established in [

22]:

where

. Hence one sufficient condition for (4.61) is

Now such an

condition (4.64) is quite standard in literature like [

8,

23], because in case

, (4.64) reads as the classical condition

In order to overcome the technical assumption (4.60) on positiveness of
, one observation is as follows.

Lemma 4.13 (Torres [8, Theorem
]).

Let
. Suppose that all gaps of consecutive zeros of all nonzero solutions
of (2.1) are strictly greater than the period 1. Then the Green function
has a constant sign.

By Theorem 4.1 of this paper, one sees that the hypothesis in Lemma 4.13 on solutions of (2.1) can yield MP or AMP. Combining ideas from [8, 9, 22], Cabada and Cid have overcome the positiveness condition (4.60) to obtain the following criteria.

Lemma 4.14 (Cabada and Cid [7, Theorem
]).

Suppose that

satisfies the following two conditions:

Then
admits AMP.

Very recently, Cabada et al. [

24,

25] have generalized criteria (4.66)-(4.67) for

to AMP of the periodic solutions of the so-called

-Laplacian problem

with the constants
being replaced by more general Sobolev constants [26].

We end the paper with some remarks.

- (i)
Recall the following trivial upper bound:

See, for example, [

26]. Criteria (4.66)-(4.67) can be deduced from Theorem 1.1 with the help of estimates (4.63) and (4.69). In fact, by Theorem 4.11, conditions (4.66) and (4.67) guarantee that

admits SAMP. For AMP of

, condition (4.67) can be improved as

Theorem 1.1 shows that condition (4.61) is optimal, while the complete generalization of condition (4.60) is

.

- (ii)
It is also possible to construct many potentials

for which

admits AMP, while (4.70) is violated. For example, let

and

be defined by

Then

and the Riemann-Lebesgue lemma shows that

in

, where

is arbitrarily fixed. In particular, it follows from Lemma 2.5 that

we conclude that for

with

,

admits AMP. However, when

is large and

,

is also large. Hence

does not satisfy (4.70).

- (iii)