1 Introduction and Main Results

Let w(z)be meromorphic in the complex plane,and we use the standard notations of Nevanlinna theory of meromorphic function(see[1]–[2]),which are also introduced as follows for convenient to read:

where n(r,w)counts the number of the pole of w(z)in|z|≤r,each pole according to its multiplicity,and(r,w)counts the number of distinct poles of w(z)in|z|≤r.We call an error term and denote by S(r,w)any quantity satisfying

as r→∞,possibly outside a set of r of finite linear measure.

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Some authors investigated the existence of meromorphic solutions of algebraic differential equations and obtained many meaningful results(see[3]–[11]).Gackstatter and Laine[3]considered the following differential equation

They gave the conjecture that:the differential equation(1.1)does not possess any admissible solutions if 1≤m≤n−1.For simplicity,we denotew is called an admissible solution of(1.1)if S(r)=o(T(r,w)).

He and Laine[6]proved the conjecture in[3].While they obtained the following theorem:

Theorem A[6]The differential equation(1.1)does not possess any admissible solutions for 1≤m≤n−1.

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Gao[7]investigated the following high-order differential equation:

where

and

He obtained the result as below:

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Theorem B[7] Let w(z)be a meromorphic solution of(1.2).If

then w(z)is a non-admissible solution.

Remark 1.1 From the proof of Theorem B,we know that p−s−＞0,if p＜m.Therefore,the condition p≥m can be removed(see[7]).

Motivated by[7],we reinvestigate meromorphic solution of(1.2)when

More precisely,one of our results can be stated as follows:

where

then w(z)is a non-admissible solution.

Clearly,Theorem 1.1 reduces to Theorem B when λ=0.

Example 1.1 Obviously,w(z)=ezis an admissible solution of the following differential equation:

It is easy to see that

and

but

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Example 1.2 Clearly,w(z)=ezis the non-admissible solution of following differential equation:

here

and we have

Examples 1.1 and 1.2 imply that the conditions in Theorem 1.1 are necessary and sharp.

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We mention for the existence of solutions,growth of solutions or the form of systems of differential equations(see[12]–[17]).Particularly,Su and Li[15]considered the following systems of complex differential equation:

Theorem 1.1 Let w(z)be a meromorphic solution of(1.2).If

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and

They proved the existence of non-admissible solutions of the above equations.

In this paper,we mainly concern the following systems of nonlinear complex algebraic differential equations:

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Lemma 2.3[10]Let w,a0,a1,···,akbe meromorphic functions in|z|＜ ∞ and T(r,aj)=S(r,w)(j=1,2,···,k).Then

and

are differential polynomials with meromorphic coefficients{ck(z)},{dl(z)},

and

are differential monomials.

Now we introduce two examples to show that some conditions in Theorem 1.2 are necessary.

One of main results can be stated as below:

Theorem 1.2 Let(w1,w2)be a meromorphic solution of(1.3).If p＜m2,q＜m1,s≤p−2,t≤q−2,and

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then(w1,w2)is a non-admissible meromorphic solution.

Definition 1.1 Let(w1(z),w2(z))be a meromorphic solution of(1.3),S(r)be the sum of all characteristic functions of coefficients in(1.3).If wk(z)satis fies S(r)=o(T(r,wk(z))(k=1,2)outside a possible exception set E with finite linear measure,we call that(w1,w2)is an admissible solution,or else,non-admissible solution.

Example 1.3 Consider the following systems of differential equations:

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where(w1,w2)=(e2z,ez)is a solution of the differential equations containing two nonadmissible components,and

Obviously,we have

and

Example 1.4 is an admissible solution of the following equations:

which satis fies

One can easily see that

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but

Generally,we have

Theorem 1.3 Let(w1,w2)be a meromorphic solution of(1.3).If

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and

where λ is a constant with λ ＜ 1,T(r)=min{T(r,w1),T(r,w2)},and I is a subset of(0,+∞)with in finite linear measure,then(w1,w2)is a non-admissible meromorphic solution.

In fact,Theorem 1.2 is a special case of Theorem 1.3,that is λ=0.

In this paper,We also discuss the following systems of differential equations:

where

and

are differential monomials,ap(z) 0,bq(z) 0,as(z) 0,bt(z) 0,{ai(z)},{bj(z)}are meromorphic coefficients.Concerning the existence of meromorphic solution,we have

Theorem 1.4 Let(w1,w2)be a meromorphic solution of(1.4).If p＜m2,q＜m1,s≤p−2,t≤q−2 and m2m1,then(w1,w2)is a non-admissible meromorphic solution.

Corollary 1.1 Let(w1,w2)be a meromorphic solution of(1.4).If p＜m2,q＜m1,s＜p−2,t＜q−2 and m2=m1,then(w1,w2)is a non-admissible solution.

Based on Gao’s results(see[11]and[17]),we apply Theorem 1.2 to discuss(1.4).Precisely,we get

Theorem 1.5 Let(w1,w2)be an admissible meromorphic solution of(1.3).If p＜m2,q＜m1,and

then(1.3)takes one of the following forms:

where

2 Preliminaries

In this section,we introduce some lemmas.They would bring a great convenience during our analysis.Namely,

Lemma 2.1[4]Let w,b0,b1,···,bkbe meromorphic functions in|z| ＜ ∞,bk 0,then there hold:

Lemma 2.2[5]Let g0(z)and g1(z)be meromorphic functions in|z|＜∞and linearly independent over C.Set g0(z)+g1(z)=Φ.Then we have

where

where p ＞ s,q ＞ t,{(k)},{(l)}are two finite index sets,ap(z) 0,bq(z) 0,as(z) 0,bt(z)0,ai(z)and bj(z)are meromorphic functions,

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If(w1,w2)is an admissible solution,then A1,B1and A2,B2are linearly independent respectively.

Then

Lemma 2.4 Let(w1,w2)be a meromorphic solution of(1.3).Putting

The proof is based on the definition of linear independence,we omit the detail here.

Lemma 2.5 Let(w1,w2)be a meromorphic admissible solution of(1.4).If p＜m2,q＜m1,then

Proof. Let(w1,w2)be a meromorphic admissible solution of(1.4).By Lemma 2.1,we get

That is,

Similarly,we have

By(2.1)and(2.2),we gain

Since m1−q＞0,m2−p＞0,we conclude

We finish the proof of the lemma.

3 Proof of Theorem 1.1

Firstly,let w(z)be an admissible meromorphic solution of the differential equation(1.2).Denote

Analogous to Lemma 2.4,we maybe easily prove that A and B are linearly independent.Meanwhile,we can see from Lemma 2.2 that

Now,we estimate very term on the right-hand side of(3.1).By using Lemmas 2.1 and 2.3,we get

Since

that is,

and since

by(3.3),we have

Thus

Inserting(3.2),(3.4)–(3.9)into(3.1),we obtain

Since

we conclude

Noting that

we have

which is a contradiction.Therefore,w(z)is not an admissible solutions of(1.2)and the proof of the theorem is completed.

4 Proof of Theorem 1.3

Suppose that(w1,w2)is a meromorphic admissible solution of the systems of differential equations(1.3).In view of Lemmas 2.2 and 2.4,we have

Applying Lemmas 2.1 and 2.3,we estimate every term in(4.1).Since

then

Meanwhile,since

then we have

and

Therefore

Inserting(4.2)–(4.9)into(4.1),we obtain

If T(r,w1)≤T(r,w2),then

Otherwise,if T(r,w1)≥T(r,w2),then

Meanwhile,noting that

we obtain

Proceeding similarly as the above proof,we can conclude

In view of p＜m2,q＜m1,we get

and

Meanwhile,since s≤p−2,t≤q−2 and 0≤λ＜1,we have one of the inequalities(4.10)and(4.11)does not always hold.This shows that our supposition is impossible.The proof of Theorem 1.3 is completed.

5 Proof of Theorem 1.4

We suppose that(w1,w2)is a meromorphic admissible solution of(1.4).Let

Then

Moreover,A1,C1and A2,C2are linearly independent respectively by Lemma 2.4.

Similarly to the proceeding of the analysis of Theorem 1.3 and together with the relations

we can derive that

That is,

Similarly,we can get

Using Lemma 2.5,we can immediately deduce that

Noting that p＜m2,q＜m1,one can see from(5.1)and(5.2)that

and

It is impossible.Because m2m1,at least one is a positive constant betweenandThis ends the proof of Theorem 1.4.

6 Proof of Theorem 1.5

We assume that the systems of differential equations(1.3)does not have the forms(1.5),(1.6)or(1.7),and admits an admissible solution(w1,w2).We can rewrite equation(1.3)as

where

By Theorem 1.2,we know that the systems of differential equations(6.1)has no admissible solution,which is a contradiction with the condition of Theorem 1.5.Hence,we complete the proof of Theorem 1.5.

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