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Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2

  • Yi Hui Xu , Yan Fang Li , Xiao Lan Liu and Hong Yan Xu EMAIL logo
Published/Copyright: November 20, 2023
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Open Mathematics
From the journal Open Mathematics Volume 21 Issue 1

Abstract

Our purpose in this article is to describe the solutions of several product-type nonlinear partial differential equations (PDEs)

( a 1 u + b 1 u z 1 + c 1 u z 2 ) ( a 2 u + b 2 u z 1 + c 2 u z 2 ) = 1 ,

and

( a 1 u + b 1 u z 1 + c 1 u z 2 ) ( a 2 u + b 2 u z 1 + c 2 u z 2 ) = e g ,

where g ( z ) is a nonconstant polynomial and a j , b j , and c j ( j = 1 , 2 ) are constants in C . The finite-order transcendental entire solution u of the first equation is of the following forms:

u ( z 1 , z 2 ) = ± 1 a 1 a 2 + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] ,

or

u ( z 1 , z 2 ) = 1 2 a 1 e Q ( z 1 , z 2 ) + 1 2 a 2 e − Q ( z 1 , z 2 ) + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] ,

where D = b 1 c 2 − b 2 c 1 , η 0 ∈ C − { 0 } , and

Q ( z 1 , z 2 ) = − 1 D [ ( a 1 c 2 + a 2 c 1 ) z 1 − ( a 1 b 2 + a 2 b 1 ) z 2 ] + η 1 , η 1 ∈ C .

The description of the forms of the solutions for these PDEs demonstrates that our results are some improvements of the previous results given by Liu, Cao, and Xu [L. Xu and T. B. Cao, Solutions of complex Fermat-type partial difference and differential-difference equations, Mediterr. J. Math. 15 (2018), 227], and [K. Liu and T. B. Cao, Entire solutions of Fermat type difference differential equations, Electron. J. Diff. Equ. 2013 (2013), No. 59, 1–10.]. Meantime, we list some examples to explain that the forms of solutions of our theorems are precise to some extent.

Keywords: transcendental; entire solution; partial differential equation; Nevanlinna theory
MSC 2010: 30D35; 35M30; 39A45

1 Introduction

In 1995, Khavinson [1] studied the eikonal (eiconal) equation in C 2

(1.1) ( u z 1 ) 2 + ( u z 2 ) 2 = 1

and obtained that any entire solution of equation (1.1) must be linear of the form u = c 1 z 1 + c 2 z 2 + c 0 , where c 1 2 + c 2 2 = 1 . This result can also be found in the study by Li [2]. In fact, equation (1.1) can be seen as a typical partial differential equation (PDE). In 2005, Li [3] discussed a nonlinear PDE with more general form and obtained the following theorem.

Theorem A

[3] Let g be a polynomial in C 2 . Then, u is an entire solution of the PDE

(1.2) ( u z 1 ) 2 + ( u z 2 ) 2 = e g

in C 2 if and only if

  1. u = f ( c 1 z 1 + c 2 z 2 ) or

  2. u = ϕ 1 ( z 1 + i z 2 ) + ϕ 2 ( z 1 − i z 2 ) ,

where f is an entire function in C satisfying

f ′ ( c 1 z 1 + c 2 z 2 ) = ± e g ⁄ 2 ,

c 1 and c 2 are two constants satisfying c 1 2 + c 2 2 = 1 , and ϕ 1 and ϕ 2 are entire functions in C satisfying

ϕ 1 ′ ( z 1 + i z 2 ) ϕ 2 ′ ( z 1 − i z 2 ) = 1 4 e g .

Moreover, the forms of f , ϕ 1 , and ϕ 2 can be found in the study by Li [4].

In general, equation (1.2) in the real variable case always arises in geometrical optics and wave propagation, which can describe the wave fronts of light in an inhomogeneous medium with a variable index of refraction e g . As is all known, PDEs have widely penetrated into many fields, such as financial mathematics, fluid mechanics, nonlinear acoustics, molecular diffusion theory in chemistry, gas dynamics, and engineering (see [5,6]). It is not usually a very simple problem to find the special entire and meromorphic solutions for a nonlinear PDE. Nevanlinna theory of meromorphic function is an important tool in studying the properties of solutions of (partial) differential equations, (partial) difference equations, function equations with several variables, and holomorphic curves, which has developed rapidly [7–9] in recent years. It is noteworthy that the method of the classic complex analyzis and Nevanlinna theory have been used in studying the solutions of a great number of PDEs and their deformations, which can be found in the studies of Cao and Xu [10] and Li [11,12]. For example, Saleeby [13–15] analyzed the entire and meromorphic solutions of some nonlinear PDEs including u 2 + ( u z 1 ) 2 + ( u z 2 ) 2 = 1 and λ u k + ∑ i = 1 n u z i m = 1 . Hu and Yang [16] investigated the forms of meromorphic solutions of a class of nonlinear PDE, and studied the growth and uniqueness of the solutions. Yuan et al. [17] obtained all traveling meromorphic exact solutions of two nonlinear physical equation by employing the method of complex analyzis. Lü [18] described the entire and meromorphic solutions of equation u t − C u m u x = Q , where C ( ≠ 0 ) is a constant, m ∈ N + , and Q is a polynomial. Xu and Cao [19,20] studied the Fermat-type PDE u 2 + ( u z 1 ) 2 = 1 and pointed out that any transcendental entire solution with finite-order of this equation has the form of u ( z 1 , z 2 ) = sin ( z 1 + g ( z 2 ) ) , where g ( z 2 ) is a polynomial in one variable z 2 . Xu and his colleagues [21–24] gave the descriptions of transcendental entire solutions of several complex systems of the partial differential difference equations in C 2 .

As being remarked by Khavinson [1] and Li [25], under the linear transformation z 1 = x + i y and z 2 = x − i y , equation (1.2) can be reduced to

(1.3) U x U y = P ,

where U ( x , y ) = u ( z 1 , z 2 ) and P ( x , y ) = e g . By a simple calculation, it yields

(1.4) A ( U x x U y y − ( U x y ) 2 ) + B U x y + C = 0 ,

where A = U x U y , B = U x P y + U y P x , and C = P x P y . Obviously, equation (1.4) can be said as a non-degenerate Monge-Ampère equation, which is widely used in differential geometry, variational methods, optimization problems, and transmission problems. There is an enormous literature dedicated to the study of Monge-Ampère equations. In view of this remark of Khavinson [1] and Li [25], there are some references focusing on the solutions of some first-order nonlinear PDEs with product form (see [26–28]).

Theorem B

[28, Theorem 1] Let g be a polynomial in C 2 , and let m be a non-negative integer. Then, u is an entire solution of the PDE u x u y = x m e g in C 2 if and only if the following assertions hold:

  1. u = ϕ 1 ( x ) + ϕ 2 ( y ) , where ϕ 1 ′ ( x ) = x m e α ( x ) and ϕ 2 ′ ( y ) = e β ( y ) satisfying α ( x ) + β ( y ) = g ( x , y ) ;

  2. u = F ( y + A x m + 1 ) , where A is a non-zero constant and ( m + 1 ) A F ′ 2 ( y + A x m + 1 ) = e g ;

  3. u = ( x k + 1 ⁄ ( k + 1 ) ) e a y + b + C , where ( a ⁄ ( k + 1 ) ) e 2 ( a y + b ) = e g , m = 2 k + 1 , and a ( ≠ 0 ) , b , C are constant.

Chen and Han, in their recent study [26], investigated the entire solutions for a series of product-type nonlinear PDEs, and obtained the following theorem.

Theorem C

[26, Theorem 1.1] Let p ( z , w ) ≠ 0 be a polynomial in C 2 , and let l ≥ 0 and m , n ≥ 1 be integers. u ( z , w ) in C 2 is an entire solution to the nonlinear first-order PDE

(1.5) ( u l u z ) m ( u l u w ) n = p ( z , w )

if and only if one of the following situations occurs:

  1. l = 0 and p ( z , w ) = q m ( z ) r n ( w ) for some nonzero polynomials q ( z ) , r ( w ) in C , and u ( z , w ) = c 1 ∫ q ( z ) d z + c 2 ∫ r ( w ) d w + c 0 for some constants c 0 , c 1 , and c 2 satisfying c 1 m c 2 n = 1 ; in particular, when p ( z , w ) = K for a constant K ( ≠ 0 ) , then u ( z , w ) is affine.

  2. l ≥ 0 and u ( z , w ) = { ( l + 1 ) c 1 ∫ q ( z , w ) d z + c 2 ∫ r ( z , w ) d w − c 1 ∫ ∫ q w ( z , w ) d z d w } 1 l + 1 for some constants c 1 and c 2 with c 1 m c 2 n = 1 , where q ( z , w ) and r ( z , w ) are nonzero polynomials in C 2 such that c 1 q w ( z , w ) = c 2 r z ( z , w ) ≢ 0 and p ( z , w ) = q m ( z , w ) r n ( z , w ) .

Theorem D

[26, Theorem 1.4] Assume that g ( z , w ) is a polynomial in C 2 , p ( z , w ) is an irreducible nonzero polynomial in C 2 with p z ( z , w ) p w ( z , w ) ≢ 0 , and l ≥ 0 and n ≥ 1 are integers. u ( z , w ) in C 2 is an entire solution to the nonlinear first-order PDE

(1.6) ( u l u z ) ( u l u w ) n = p ( z , w ) e g ( z , w )

if and only if l = 0 and u ( z , w ) = ∫ b 0 ( z ) e a 0 ( z ) d z + e − κ e a 0 ( z ) w + c 0 for some constants c 0 and κ , where a 0 ( z ) is a polynomial of z ∈ C satisfying a 0 ( z ) = g ( z , w ) + n κ n + 1 and b 0 ( z ) is a polynomial of z ∈ C (together with a 0 ( z ) ) satisfying b 0 ( z ) + e − κ a 0 ′ ( z ) w = p ( z , w ) .

From the above results, the following question can be natural raised

Question 1.1

What will be happened if u x and u y are replaced by the forms of linear combination u , u x , and u y for equations (1.3), (1.5), and (1.6)?

Motivated by Question 1.1 and the ideas of Chang and Li [29], Li [3], and Lü [28], the purpose of this article is to further investigate the solutions of the product-type nonlinear PDE. More precisely, this article mainly gives a description of some characterization of the transcendental entire solutions of the following product-type PDEs:

(1.7) ( a 1 u + b 1 u z 1 + c 1 u z 2 ) ( a 2 u + b 2 u z 1 + c 2 u z 2 ) = 1 ,

and

(1.8) ( a 1 u + b 1 u z 1 + c 1 u z 2 ) ( a 2 u + b 2 u z 1 + c 2 u z 2 ) = e g ,

where g ( z 1 , z 2 ) = α 1 z 1 + α 2 z 2 + β 0 and a j , b j , and c j ( j = 1 , 2 ) are nonzero constants in C . Obviously, equations (1.7) and (1.8) are the deformations of equations (1.3)–(1.5). Denote A = a 1 b 1 c 1 a 2 b 2 c 2 . If the rank R ( A ) of matrix A is equal to 1, i.e., a 2 a 1 = b 2 b 1 = c 2 c 1 = κ , then equation (1.8) can be equivalent to

(1.9) a 1 u + b 1 u z 1 + c 1 u z 2 = e g ˜ ⁄ 2 ,

where κ e g ˜ ⁄ 2 = e g ⁄ 2 . If 2 a 1 + α 1 b 1 + α 2 c 1 ≠ 0 , it follows that the transcendental entire solutions of equation (1.9) are of the form

u = ψ z 2 − c 1 b 1 z 1 e − a 1 b 1 z 1 + 2 2 a 1 + α 1 b 1 + α 2 c 1 e g ˜ ⁄ 2 ,

and if 2 a 1 + α 1 b 1 + α 2 c 1 = 0 , it yields

u = ψ z 2 − c 1 b 1 z 1 e − a 1 b 1 z 1 + z 1 b 1 e g ˜ ⁄ 2 ,

where ψ ( x ) is an entire function in x in C . Hence, we only deal with the case R ( A ) = 2 for equation (1.8) in this article.

In this article, we will employ the main theory tool including the Nevanlinna theory with several complex variables and the characteriztic equations for quasi-linear PDEs. Now, let us describe the basic structure of our article briefly. In Section 2, we will exhibit our main results about the forms of solutions of equations (1.7) and (1.8) and some variations list a number of examples to explain the correctness of the forms of solutions for every case and the significant difference about the order of solutions from a single variable to several variables. We will give the details of the proofs of the main results (Theorems 2.1 and 2.5) in Sections 3 and 4, respectively.

2 Results and examples

Throughout our manuscript, let us assume that β 1 , β 2 , η , η 0 , η 1 , and η 2 are constants, which can be different at each time occurrence. Let D ≔ b 1 c 2 − b 2 c 1 , D 1 ≔ a 1 b 2 − a 2 b 1 , and D 2 ≔ a 1 c 2 − a 2 c 1 . Now, we state the first main theorem which is about the existence and forms of the solutions for equation (1.7).

Theorem 2.1

If D D 1 D 2 ≠ 0 , and assume that u ( z 1 , z 2 ) is the finite-order transcendental entire solution of equation (1.7). Then, u must be one of the following cases:

  1. u ( z 1 , z 2 ) = ± 1 a 1 a 2 + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] ;

  2. u ( z 1 , z 2 ) = 1 2 a 1 e Q ( z 1 , z 2 ) + 1 2 a 2 e − Q ( z 1 , z 2 ) + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] ,

    where η 0 ∈ C − { 0 } , and

    Q ( z 1 , z 2 ) = − 1 D [ ( a 1 c 2 + a 2 c 1 ) z 1 − ( a 1 b 2 + a 2 b 1 ) z 2 ] + η 1 , η 1 ∈ C .

The following examples show the existence of transcendental entire solutions of equation (1.7) for every case in Theorem 2.1.

Example 2.1

Let η 0 ∈ C − { 0 } and

u ( z 1 , z 2 ) = ± 1 2 + η 0 e 4 z 1 − 3 z 2 .

Thus, we have that u is a transcendental entire solution of equation (1.7) with ρ ( u ) = 1 for the case a 1 = 1 , b 1 = 2 , c 1 = 3 , a 2 = 2 , b 2 = 1 , and c 2 = 2 . This example means that the form of solution of the case (i) in Theorem 2.1 is precise.

Remark 2.1

[19] The order of a meromorphic function u in C n is defined as follows:

ρ ( u ) = limsup r → ∞ log + T ( r , u ) log r ,

where T ( r , u ) is the Nevanlinna characteriztic function of u and log + x = max { 0 , log x } for x > 0 .

Example 2.2

Let η 0 ∈ C and

u ( z 1 , z 2 ) = 1 4 e − 9 z 1 + 7 z 2 + 1 2 e 9 z 1 − 7 z 2 + η 0 e − 7 z 1 + 5 z 2 .

Thus, we have that u ( z 1 , z 2 ) is a transcendental entire solution of equation (1.7) with ρ ( u ) = 1 for the case a 1 = 2 , b 1 = 1 , c 1 = 1 , a 2 = 1 , b 2 = 3 , and c 2 = 4 . Obviously, this example corresponds to the case (ii) in Theorem 2.1.

From Theorem 2.1, one can obtain the following corollary easily:

Corollary 2.1

The finite-order transcendental entire solution u ( z 1 , z 2 ) of equation

(2.1) ( u + u z 1 ) ( u + u z 2 ) = 1

in C 2 is of the following forms:

u ( z 1 , z 2 ) = ± 1 + η 0 e − ( z 1 + z 2 ) ,

or

u ( z 1 , z 2 ) = e z 1 − z 2 + β 0 + e − z 1 + z 2 − β 0 + η 0 e − ( z 1 + z 2 ) ,

where β 0 , η 0 ∈ C .

When a 1 = a 2 = 0 , i.e., D 1 = 0 in equation (1.7), similar to the argument as in Theorem 2.1, we have the following theorem.

Theorem 2.2

Let D ≠ 0 and u ( z 1 , z 2 ) be the entire solution of

( b 1 u z 1 + c 1 u z 2 ) ( b 2 u z 1 + c 2 u z 2 ) = 1 .

Then,

u = d ξ − b ξ − 1 D z 1 + a ξ − 1 − c ξ D z 2 + ξ 0 ,

where ξ ( ≠ 0 ) , ξ 0 ∈ C .

For the case D = 0 , we study the solutions of the following PDEs:

(2.2) ( a 1 u + b 1 u z 1 ) ( a 2 u + b 2 u z 1 ) = 1 ,

(2.3) [ a 1 u + b 1 ( u z 1 + u z 2 ) ] [ a 2 u + b 2 ( u z 1 + u z 2 ) ] = 1 ,

and obtain the following theorem.

Theorem 2.3

Let a j , b j ∈ C − { 0 } , j = 1 , 2 , and D 1 ≠ 0 . If equation (2.2) admits the transcendental entire solution u ( z 1 , z 2 ) , then a 1 b 2 + a 2 b 1 = 0 and u ( z 1 , z 2 ) satisfies the following form:

u ( z 1 , z 2 ) = 1 2 a 1 e a 1 b 1 z 1 + ϕ 1 ( z 2 ) + 1 2 a 2 e − a 1 b 1 z 1 − ϕ 1 ( z 2 ) ,

where ϕ 1 ( z 2 ) is an entire function in z 2 .

Theorem 2.4

Let a j , b j ∈ C − { 0 } , j = 1 , 2 , and D 1 ≠ 0 . If u ( z 1 , z 2 ) is a transcendental entire solution of equation (2.3), then a 1 b 2 + a 2 b 1 = 0 and u ( z 1 , z 2 ) must satisfy the following form:

u ( z 1 , z 2 ) = 1 2 a 1 e a 1 b 1 z 1 + ϕ 2 ( z 2 − z 1 ) + 1 2 a 2 e − a 1 b 1 z 1 − ϕ 2 ( z 2 − z 1 ) ,

where ϕ 2 ( z 2 − z 1 ) is an entire function in z 2 − z 1 .

Two examples explain the existence of transcendental entire solutions of equation (2.2) and (2.3).

Example 2.3

Let

u ( z 1 , z 2 ) = 1 4 e 2 3 z 1 + z 2 2 + 1 4 e − 2 3 z 1 − z 2 2 .

Thus, u ( z 1 , z 2 ) is a transcendental entire solution of equation (2.2) with ρ ( u ) = 2 for the case a 1 = 2 , b 1 = 3 , a 2 = 2 , and b 2 = − 3 .

Example 2.4

Let

u ( z 1 , z 2 ) = 1 2 e 1 2 z 1 + ( z 2 − z 1 ) 3 + 1 2 e − 1 2 z 1 − ( z 2 − z 1 ) 3 .

Thus, u is a transcendental entire solution of equation (2.2) with ρ ( u ) = 3 for the case a 1 = 1 , b 1 = 2 , a 2 = 1 , and b 2 = − 2 .

Before stating the result of equation (1.8), let us consider the following notations. Denote

A 11 ≔ 1 D [ ( a 1 c 2 + a 2 c 1 ) + c 2 ( b 1 α 1 + c 1 α 2 ) ] , A 12 ≔ − 1 D [ ( a 1 b 2 + a 2 b 1 ) + b 2 ( b 1 α 1 + c 1 α 2 ) ] , A 21 ≔ − 1 D [ ( a 1 c 2 + a 2 c 1 ) + c 1 ( b 2 α 1 + c 2 α 2 ) ] , A 22 ≔ 1 D [ ( a 1 b 2 + a 2 b 1 ) + b 1 ( b 2 α 1 + c 2 α 2 ) ] .

Obviously, we have

(2.4) A 11 + A 21 = α 1 , A 12 + A 22 = α 2 , g ( z ) = ( A 11 + A 21 ) z 1 + ( A 12 + A 22 ) z 2 + β 0 .

Theorem 2.5

If D D 1 D 2 ≠ 0 , and let g ( z 1 , z 2 ) = α 1 z 1 + α 2 z 2 + β 0 , α 1 , α 2 , β 0 ∈ C . If u ( z 1 , z 2 ) is a transcendental entire solution of equation (1.8) with finite-order, then u must be one of the following cases:

  1. If μ 1 μ 2 ≠ 0 , then

    u ( z 1 , z 2 ) = ± 2 μ 1 μ 2 e g ⁄ 2 + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − b 1 a 2 ) z 2 ] ,

    where η 0 is a constant, μ 1 = 2 a 1 + b 1 α 1 + c 1 α 2 and μ 2 = 2 a 2 + b 2 α 1 + c 2 α 2 ;

  2. If μ 1 μ 2 ≠ 0 , then

    u ( z 1 , z 2 ) = 1 μ 1 e p ( z ) − 1 μ 2 e q ( z ) + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] ;

    if μ 1 = 0 and μ 2 ≠ 0 or μ 1 ≠ 0 and μ 2 = 0 , then

    u ( z 1 , z 2 ) = η 1 z 1 e 1 2 ( α 1 z 1 + α 2 z 2 ) + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] ;

    if μ 1 = 0 and μ 2 = 0 , i.e., α 1 = 2 ( a 2 c 1 − a 1 c 2 ) D and α 2 = 2 ( a 1 b 2 − a 2 b 1 ) D , then

    u ( z 1 , z 2 ) = ( η 1 z 1 + η 0 ) e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] ;

    where p ( z ) ≠ q ( z ) , and

    p ( z ) = A 11 z 1 + A 12 z 2 + A 10 , q ( z ) = A 21 z 1 + A 22 z 2 + A 20 ,

    A 10 + A 20 = β 0 , and η 0 and η 1 are constants such that b 1 b 2 η 1 2 = e β 0 , η 0 ∈ C .

Remark 2.2

Examples 2.5–2.8 show that the forms of solutions in Theorem 2.5 are evident different with that in Theorem B. Moreover, they cannot be included in each other.

The following examples show that the forms of the finite-order transcendental entire solutions of equation (1.8) are precise.

Example 2.5

Let

u ( z ) = 2 21 e z 1 − 1 2 z 2 + η 0 e − 1 3 ( 2 z 1 + z 2 ) .

Thus, u is a transcendental entire solution of equation (1.8) with ρ ( u ) = 1 for the case a 1 = 1 , b 1 = 2 , c 1 = − 1 , a 2 = b 2 = c 2 = 1 , and g ( z ) = 2 z 1 − z 2 . This example shows that the form of solution for the case in Theorem 2.5 (i) is precise.

Example 2.6

Let η 0 ∈ C and

u ( z 1 , z 2 ) = 1 8 e 6 z 1 − 5 z 2 − 1 2 e − 4 z 1 + 7 z 2 + η 0 e − 2 z 1 + 3 z 2 .

Thus, u is a transcendental entire solution of equation (1.8) with ρ ( u ) = 1 for the case a 1 = 1 , b 1 = 2 , c 1 = 1 , a 2 = − 1 , b 2 = c 2 = 1 , and g ( z ) = 2 z 1 + 2 z 2 . Indeed, we have μ 1 = 8 ≠ 0 , and μ 2 = 2 ≠ 0 by a simple calculation. So, this example shows that the form in the case μ 1 μ 2 ≠ 0 in Theorem 2.5 is precise.

Example 2.7

Let η 0 ∈ C , e β 0 = 6 , and

u ( z 1 , z 2 ) = z 1 e 2 z 1 − 5 z 2 + η 0 e 2 z 1 + 5 z 2 .

Thus, u is a transcendental entire solution of equation (1.8) with ρ ( u ) = 1 for the case a 1 = 1 , b 1 = 2 , c 1 = 1 , a 2 = − 1 , b 2 = 3 , c 2 = 1 , and g ( z ) = 4 z 1 − 10 z 2 + β 0 . Indeed, we have μ 2 = 0 and μ 1 = 3 ≠ 0 by a simple calculation. So, this example shows that the form in the case μ 2 = 0 and μ 1 ≠ 0 in Theorem 2.5 is precise.

Example 2.8

Let

u ( z 1 , z 2 ) = ( z 1 + 2 ) e 3 z 1 − 2 z 2 .

Thus, u is a transcendental entire solution of equation (1.8) with ρ ( u ) = 1 for the case a 1 = 1 , b 1 = 1 , c 1 = 2 , a 2 = − 1 , b 2 = 1 , c 2 = 1 , and g ( z ) = 6 z 1 − 4 z 2 .

From Theorem 2.5, one can obtain the following corollary easily.

Corollary 2.2

Let g ( z 1 , z 2 ) = α 1 z 1 + α 2 z 2 + β 0 , α 1 , α 2 , β 0 ∈ C , and u ( z 1 , z 2 ) be a finite-order transcendental entire solution of equation

( u + u z 1 ) ( u + u z 2 ) = e g .

  1. If ( α 1 + 2 ) ( α 2 + 2 ) ≠ 0 , then

    u ( z 1 , z 2 ) = ± 2 e g ⁄ 2 ( α 1 + 2 ) ( α 2 + 2 ) + η 0 e − ( z 1 + z 2 ) ;

  2. If ( α 1 + 2 ) ( α 2 + 2 ) ≠ 0 , then

    u ( z 1 , z 2 ) = 1 α 1 + 2 e ( α 1 + 1 ) z 1 − z 2 + β 1 + 1 α 2 + 2 e − z 1 + ( α 1 + 1 ) z 2 + β 2 + η 0 e − ( z 1 + z 2 ) ;

    if α 1 = − 2 , α 2 ≠ − 2 , then

    u ( z 1 , z 2 ) = 1 α 2 + 2 e − z 1 + ( α 2 + 1 ) z 2 + β 2 + ( η 1 z 1 + η 0 ) e − ( z 1 + z 2 ) ;

    if α 1 ≠ − 2 , α 2 = − 2 , then

    u ( z 1 , z 2 ) = 1 α 1 + 2 e ( α 1 + 1 ) z 1 − z 2 + β 1 + ( η 2 z 1 + η 0 ) e − ( z 1 + z 2 ) ;

    if α 1 = − 2 , α 2 = − 2 , then

    u ( z 1 , z 2 ) = [ η 3 ( z 1 + z 2 ) + η 0 ] e − ( z 1 + z 2 ) ,

    where β 1 , β 2 , η , η 0 , η 1 , η 2 , and η 3 are constants such that β 1 + β 2 = β 0 , η 1 = e β 0 − β 2 , η 2 = e β 0 − β 1 , and η 3 2 = e β 0 .

Corresponding for Theorems 2.3 and 2.4, similar to the argument as in the proof of Theorem 2.5, we have the following theorem.

Theorem 2.6

Let D 1 ≠ 0 and g ( z 1 , z 2 ) = α 1 z 1 + α 2 z 2 + β 0 , α 1 , α 2 , β 0 ∈ C . If equation

(2.5) ( a 1 u + b 1 u z 1 ) ( a 2 u + b 2 u z 1 ) = e g

admits the transcendental entire solution u ( z 1 , z 2 ) with finite-order, then u ( z 1 , z 2 ) satisfies the following forms:

u ( z 1 , z 2 ) = ± 2 ( 2 a 1 + b 1 α 1 ) ( 2 a 2 + b 2 α 1 ) e g ⁄ 2 ,

or

u ( z 1 , z 2 ) = 1 D 1 b 2 e − a 2 b 2 z 1 + ϕ 1 ( z 2 ) − b 1 e − a 1 b 1 z 1 + ϕ 2 ( z 2 ) ,

and

a 1 b 1 + a 2 b 2 = − α 1 ,

where ϕ 1 ( z 2 ) , and ϕ 2 ( z 2 ) are polynomials in z 2 with ϕ 1 ( z 2 ) + ϕ 2 ( z 2 ) = α 2 z 2 + β 0 .

Theorem 2.7

Let D 1 ≠ 0 and g ( z 1 , z 2 ) = α 1 z 1 + α 2 z 2 + β 0 , α 1 , α 2 , β 0 ∈ C . If equation

(2.6) [ a 1 u + b 1 ( u z 1 + u z 2 ) ] [ a 2 u + b 2 ( u z 1 + u z 2 ) ] = e g

admits the transcendental entire solution u ( z 1 , z 2 ) with finite-order, then u ( z 1 , z 2 ) satisfies the following forms:

u ( z 1 , z 2 ) = ± 2 [ 2 a 1 + b 1 ( α 1 + α 2 ) ] [ 2 a 2 + b 2 ( α 1 + α 2 ) ] e g ⁄ 2 ,

or

u ( z 1 , z 2 ) = 1 D 1 b 2 e − a 2 b 2 z 1 + R 1 ( z 2 − z 1 ) − b 1 e − a 1 b 1 z 1 + R 2 ( z 2 − z 1 ) ,

and

a 1 b 1 + a 2 b 2 = − ( α 1 + α 2 ) ,

where R j ( z 2 − z 1 ) and ( j = 1 , 2 ) are two polynomials in z 2 − z 1 such that R 1 ( z 2 − z 1 ) + R 2 ( z 2 − z 1 ) = α 2 ( z 2 − z 1 ) + β 0 .

The following examples show the existence of transcendental entire solutions of equations (2.5) and (2.6).

Example 2.9

Let

u ( z 1 , z 2 ) = − 2 6 i e − 7 12 z 1 + z 2 .

Thus, u is a transcendental entire solution of equation (2.5) with ρ ( u ) = 1 for the case a 1 = 2 , b 1 = 3 , a 2 = 1 , b 2 = 1 , and g ( z ) = − 7 6 z 1 + 2 z 2 .

Example 2.10

Let

u ( z 1 , z 2 ) = e − z 1 + 4 z 2 3 − z 2 − 2 e − 3 2 z 1 − 4 z 2 3 + 3 z 2 .

Thus, u is a transcendental entire solution of equation (2.5) with ρ ( u ) = 3 for the case a 1 = 3 , b 1 = 2 , a 2 = b 2 = 1 , and g ( z ) = − 5 2 z 1 + 2 z 2 .

Example 2.11

Let

u ( z 1 , z 2 ) = 1 2 5 e 2 z 1 − z 2 .

Thus, u is a transcendental entire solution of equation (2.6) with ρ ( u ) = 1 for the case a 1 = b 1 = a 2 = 2 , b 2 = 3 , and g ( z ) = 4 z 1 − 2 z 2 .

Example 2.12

Let

u ( z 1 , z 2 ) = e − 2 z 1 + 4 ( z 2 − z 1 ) n − ( z 2 − z 1 ) − e − 3 z 1 − 4 ( z 2 − z 1 ) n − 2 ( z 2 − z 1 ) .

Thus, u is a transcendental entire solution of equation (2.6) with ρ ( u ) = n for the case a 1 = 3 , a 2 = 2 , b 1 = b 2 = 1 , and g ( z ) = − 2 z 1 − 3 z 2 .

Remark 2.3

Examples 2.10 and 2.12 show that equations (2.5) and (2.6) have the transcendental entire solutions with any integer order.

3 Proofs of Theorems 2.1–2.4

3.1 The Proof of Theorem 2.1

Proof

First, we assume that u is a finite-order transcendental entire solution of equation (2.1). Due to equation (2.1), we have that a 1 u + b 1 u z 1 + c 1 u z 2 and a 2 u + b 2 u z 1 + c 2 u z 2 have no any zero and pole. Then, there exists a polynomial h such that

(3.1) a 1 u + b 1 u z 1 + c 1 u z 2 = e h , a 2 u + b 2 u z 1 + c 2 u z 2 = e − h .

In view of D = b 1 c 2 − b 2 c 1 ≠ 0 , we can deduce from equation (3.1) that

(3.2) u z 1 = 1 D [ c 2 e h − c 1 e − h + ( a 2 c 1 − a 1 c 2 ) u ] ,

and

(3.3) u z 2 = 1 D [ b 1 e − h − b 2 e h + ( a 1 b 2 − a 2 b 1 ) u ] .

Noting that u z 1 z 2 = u z 2 z 1 , we have from equations (3.2) and (3.3) that

(3.4) ( b 2 h z 1 + c 2 h z 2 ) e h + ( b 1 h z 1 + c 1 h z 2 ) e − h = ( a 1 b 2 − a 2 b 1 ) u z 1 + ( a 1 c 2 − a 2 c 1 ) u z 2 .

Substituting equations (3.2) and (3.3) into equation (3.4), it yields that

(3.5) ( a 2 + b 2 h z 1 + c 2 h z 2 ) e 2 h = a 1 − b 1 h z 1 − c 1 h z 2 .

Now, this result is divided into two cases, as shown below.

Case 1. If h is a constant, denote e h = ξ . Then, it follows from equation (3.5) that

(3.6) a 1 ξ − 1 = a 2 ξ .

By observing equation (3.1), we can deduce that

( a 1 b 2 − a 2 b 1 ) u z 1 + ( a 1 c 2 − a 2 c 1 ) u z 2 = a 1 ξ − 1 − a 2 ξ ,

and in view of that (3.6), we have

(3.7) ( a 1 b 2 − a 2 b 1 ) u z 1 + ( a 1 c 2 − a 2 c 1 ) u z 2 = 0 ,

which implies that

(3.8) u = φ [ ( a 1 c 2 − a 2 c 1 ) z 1 − ( a 1 b 2 − a 2 b 1 ) z 2 ] ,

where φ ( x ) is a finite-order transcendental entire function of x in C .

Substituting equation (3.8) into equation (3.1), it follows that

a 1 φ + a 1 ( b 2 c 1 − b 1 c 2 ) φ ′ = ξ .

Solving this equation, we have

(3.9) φ = ξ a 1 + η 0 e 1 D x ,

where η 0 ∈ C . In view of equations (3.6), (3.8), and (3.9), we have

(3.10) u ( z 1 , z 2 ) = ± 1 a 1 a 2 + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] .

Therefore, this completes the proof of Theorem 2.1(i).

Case 2. If h is not a constant, we can obtain that h is a nonconstant polynomial in C 2 . In fact, if a 2 + b 2 h z 1 + c 2 h z 2 ≠ 0 , we have from equation (3.5) that

(3.11) e 2 h = a 1 − b 1 h z 1 − c 1 h z 2 a 2 + b 2 h z 1 + c 2 h z 2 .

If h is a nonconstant polynomial, by using the basic Nevanlinna results (see, e.g., [30, p. 99], [31] or [32, Lemma 3.2]), one can conclude from equation (3.11) that T ( r , e 2 h ) = O { T ( r , h ) + log r } , outside possibly a set of finite Lebesgue measure, where T ( r , F ) denotes the Nevanlinna characteriztic function of a meromorphic function F in C 2 . On the other hand, we have lim r → ∞ T ( r , e 2 h ) T ( r , h ) + log r = + ∞ for h being not a constant. Due to the fact that h is a polynomial, we can obtain that h must be constant. Thus, this is a contradiction. If h is a transcendental entire function, similar to the above argument, we also obtain a contradiction. Therefore, we have a 2 + b 2 h z 1 + c 2 h z 2 = 0 . This leads to a 1 − b 1 h z 1 − c 1 h z 2 = 0 . In view of D ≠ 0 , and by solving this system of equations, we have

(3.12) h z 1 = a 1 c 2 + a 2 c 1 D , h z 2 = − a 1 b 2 + a 2 b 1 D ,

which means that

(3.13) h = 1 D [ ( a 1 c 2 + a 2 c 1 ) z 1 − ( a 1 b 2 + a 2 b 1 ) z 2 ] + η 0 , η 0 ∈ C .

In view of equations (3.4) and (3.13), we obtain the following PDE

(3.14) ( a 1 b 2 − a 2 b 1 ) u z 1 + ( a 1 c 2 − a 2 c 1 ) u z 2 = a 1 e − h − a 2 e h ,

where h is stated as in equation (3.13). The characteriztic equations of equation (3.14) are

d z 1 d t = a 1 b 2 − a 2 b 1 , d z 2 d t = a 1 c 2 − a 2 c 1 , d u d t = a 1 e − h − a 2 e h .

Using the initial conditions: z 1 = 0 , z 2 = s , and u = u ( 0 , s ) = φ ( s ) with a parameter s , we obtain the following parametric representation for the solutions of the characteriztic equations: z 1 = ( a 1 b 2 − a 2 b 1 ) t , z 2 = ( a 1 c 2 − a 2 c 1 ) t + s , and

u ( t , s ) = ∫ 0 t ( a 1 e − h − a 2 e h ) d t + φ ( s ) ,

where φ ( s ) is a polynomial in s ≔ z 2 − a 1 c 2 − a 2 c 1 a 1 b 2 − a 2 b 1 z 1 . By a simple calculation, we have

(3.15) u ( z 1 , z 2 ) = 1 2 a 1 e h + 1 2 a 2 e − h + φ ( s ) .

Substituting equation (3.15) into equation (3.1), we have

φ ( s ) − b 1 c 2 − b 2 c 1 a 1 b 2 − a 2 b 1 φ ′ ( s ) = 0 ,

which leads to

(3.16) φ ( z 1 , z 2 ) = e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] + η 1 , η 1 ∈ C .

Thus, we can prove the conclusion (ii) from equations (3.13), (3.15), and (3.16).

Therefore, this completes the proof of Theorem 2.1.□

3.2 Proofs of Theorems 2.3 and 2.4

Since the proofs of Theorems 2.3 and 2.4 are very similar, we thus only give the proof of Theorem 2.4 below. Let u be the transcendental entire solution of equation (2.2), and by employing the same argument as in the proof of Theorem 2.1, there exists a polynomial h such that

(3.17) a 1 u + b 1 ( u z 1 + u z 2 ) = e h , a 2 u + b 2 ( u z 1 + u z 2 ) = e − h .

Noting that D 1 = a 1 b 2 − a 2 b 1 ≠ 0 , we have

(3.18) u = 1 D 1 ( b 2 e h − b 1 e − h ) , u z 1 + u z 2 = 1 D 1 ( a 1 e − h − a 2 e h ) .

Obviously, h cannot be a constant. Otherwise, it follows from the first equation in equation (3.18) that u is a constant. Thus, it yields from equation (3.18) that

1 D 1 ( a 1 e − h − a 2 e h ) = u z 1 + u z 2 = 1 D 1 ( b 2 e h + b 1 e − h ) ( h z 1 + h z 2 ) ,

i.e.,

(3.19) [ a 2 + b 2 ( h z 1 + h z 2 ) ] e 2 h = a 1 − b 1 ( h z 1 + h z 2 ) .

If a 2 + b 2 ( h z 1 + h z 2 ) ≠ 0 , then we have from equation (3.19) that

e 2 h = a 1 − b 1 ( h z 1 + h z 2 ) a 2 + b 2 ( h z 1 + h z 2 ) .

Similar to the same argument as in Case 2 of Theorem 2.1, we can obtain a contradiction.

If a 2 + b 2 ( h z 1 + h z 2 ) = 0 , we thus have from equation (3.19) that a 1 − b 1 ( h z 1 + h z 2 ) = 0 . These lead to

(3.20) h z 1 + h z 2 = a 1 b 1 = − a 2 b 2 .

This means that a 1 b 2 + a 2 b 1 = 0 . Solving equation (3.20), we have

(3.21) h = a 1 b 1 z 1 + ϕ 2 ( z 2 − z 1 ) ,

where ϕ 2 ( z 2 − z 1 ) is an entire function of z 2 − z 1 . By combining with equations (3.18) and (3.21), we can conclude

u = 1 2 a 1 e a 1 b 1 z 1 + ϕ 2 ( z 2 − z 1 ) + 1 2 a 2 e − a 1 b 1 z 1 − ϕ 2 ( z 2 − z 1 ) .

Therefore, this completes the proof of Theorem 2.4.

4 Proof of Theorem 2.5

Proof

Let u ( z ) be a transcendental entire solution of equation (1.8) with finite-order. Noting that the fact that g = α 1 z 1 + α 2 z 2 + β 0 and u being entire function, we have that there exist two polynomials p ( z ) and q ( z ) in C 2 such that

(4.1) p ( z ) + q ( z ) = g ( z ) ,

and

(4.2) a 1 u + b 1 u z 1 + c 1 u z 2 = e p , a 2 u + b 2 u z 1 + c 2 u z 2 = e q .

In view of D = b 1 c 2 − b 2 c 1 ≠ 0 , we can deduce from equation (4.2) that

(4.3) u z 1 = 1 D [ c 2 e p − c 1 e q + ( a 2 c 1 − a 1 c 2 ) u ] ,

and

(4.4) u z 2 = 1 D [ b 1 e q − b 2 e p + ( a 1 b 2 − a 2 b 1 ) u ] .

Noting that u z 1 z 2 = u z 2 z 1 , we can deduce from equations (4.3) and (4.4) that

(4.5) ( c 2 p z 2 + b 2 p z 1 ) e p + ( a 2 c 1 − a 1 c 2 ) u z 2 = ( b 1 q z 1 + c 1 q z 2 ) e q + ( a 1 b 2 − a 2 b 1 ) u z 1 .

Substituting equations (4.3) and (4.4) into equation (4.5), we have

(4.6) ( a 2 + b 2 p z 1 + c 2 p z 2 ) e p − q = a 1 + b 1 q z 1 + c 1 q z 2 .

Now, we consider the following two cases:

Case 1. If a 2 + b 2 p z 1 + c 2 p z 2 ≠ 0 , then equation (4.6) can be written as follows:

(4.7) e p − q = a 1 + b 1 q z 1 + c 1 q z 2 a 2 + b 2 p z 1 + c 2 p z 2 .

If p − q is not a constant, noting that p and q are polynomials, we can obtain a contradiction that the left side of equation (4.7) is transcendental and the right side of equation (4.7) is rational. Thus, p − q must be a constant. Set p − q = ξ ( Const . ) . It then follows that

(4.8) p z 1 = q z 1 , p z 2 = q z 2 .

Noting that p + q = g , we have

(4.9) p = g + ξ 2 = 1 2 ( α 1 z 1 + α 2 z 2 + β 0 + ξ ) , q = g − ξ 2 = 1 2 ( α 1 z 1 + α 2 z 2 + β 0 − ξ ) .

Thus, it yields from equations (4.7)–(4.9) that

(4.10) e ξ = 2 a 1 + b 1 α 1 + c 1 α 2 2 a 2 + b 2 α 1 + c 2 α 2 .

Equation (4.5) can be rewritten as follows:

(4.11) ( a 1 b 2 − a 2 b 1 ) u z 1 + ( a 1 c 2 − a 2 c 1 ) u z 2 = ( a 1 − a 2 e ξ ) e q .

The characteriztic equations of equation (4.11) are

d z 1 d t = a 1 b 2 − a 2 b 1 , d z 2 d t = a 1 c 2 − a 2 c 1 , d u d t = ( a 1 − a 2 e ξ ) e q .

Using the initial conditions: z 1 = 0 , z 2 = s , and u = u ( 0 , s ) = φ ( s ) with a parameter s , we obtain the following parametric representation for the solutions of the characteriztic equations: z 1 = ( a 1 b 2 − a 2 b 1 ) t , z 2 = ( a 1 c 2 − a 2 c 1 ) t + s , and

u ( t , s ) = ∫ 0 t ( a 1 − a 2 e ξ ) e q ( t , s ) d t + φ ( s ) = 2 2 a 2 + b 2 A 1 + c 2 A 2 e q ( t , s ) + φ ( s ) ,

where φ ( s ) is a finite-order entire function in s . By combining with equations (4.9) and (4.10), we have

(4.12) u ( z 1 , z 2 ) = ± 2 μ 1 μ 2 e g 2 + φ ( s ) ,

where μ 1 = 2 a 1 + b 1 α 1 + c 1 α 2 , μ 2 = 2 a 2 + b 2 α 1 + c 2 α 2 and s = z 2 − a 1 c 2 − a 2 c 1 a 1 b 2 − a 2 b 1 z 1 .

Substituting equation (4.12) into any equations in equation (4.2), we have

a 1 φ ( s ) + b 1 a 1 c 2 − a 2 c 1 a 2 b 1 − a 1 b 2 + c 1 φ ′ ( s ) = 0 ,

which leads to

(4.13) φ ( s ) = η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] .

By combining with equations (4.12) and (4.13), we can obtain the conclusion (i) of Theorem 2.5.

Case 2. If a 2 + b 2 p z 1 + c 2 p z 2 = 0 , it then follows from equation (4.6) that a 1 + b 1 q z 1 + c 1 q z 2 = 0 . Thus, we can deduce that

(4.14) p ( z 1 , z 2 ) = − a 2 b 2 z 1 + ψ 1 z 2 − c 2 b 2 z 1 ,

and

(4.15) q ( z 1 , z 2 ) = − a 1 b 1 z 1 + ψ 2 z 2 − c 1 b 1 z 1 ,

where ψ 1 and ψ 2 are two polynomials in s 1 ≔ z 2 − c 2 b 2 z 1 and s 2 ≔ z 2 − c 1 b 1 z 1 , respectively.

Now, we will prove that p and q are the linear forms of z 1 and z 2 . Set

ψ 1 ( s 1 ) = λ 1 , m s 1 m + λ 1 , m − 1 s 1 m − 1 + ⋯ + λ 1 , 1 s 1 + λ 1 , 0 ,

and

ψ 2 ( s 2 ) = λ 2 , n s 2 n + λ 2 , n − 1 s 2 n − 1 + ⋯ + λ 2 , 1 s 2 + λ 2 , 0 ,

where λ 1 , i , λ 2 , j ∈ C , i = 0 , 1 , … , j = 0 , 1 , … , n and λ 1 , m λ 1 , n ≠ 0 . Noting that p + q = g , we have

(4.16) ψ 1 ( s 1 ) + ψ 2 ( s 2 ) = α 1 + a 1 b 1 + a 2 b 2 z 1 + α 2 z 2 + β 0 .

Assume that m ≥ 2 , by observing the coefficients of z 2 m of two sides of equation (4.16), it follows that m = n and λ 1 , m = − λ 1 , n . This means that

(4.17) z 2 − c 2 b 2 z 1 m = z 2 − c 1 b 1 z 1 m .

By observing the coefficients of z 2 m − 1 z 1 of two sides of equation (4.17), we can deduce that

c 2 b 2 z 2 m − 1 z 1 = c 1 b 1 z 2 m − 1 z 1 .

This is a contradiction with D = b 1 c 2 − b 2 c 1 ≠ 0 . Hence, it yields that m = n ≤ 1 . This means that p and q are the linear forms of z 1 and z 2 . By combining with a 2 + b 2 p z 1 + c 2 p z 2 = 0 , a 1 + b 1 q z 1 + c 1 q z 2 = 0 , and p + q = g , after a simple calculation, we have

(4.18) p = A 11 z 1 + A 12 z 2 + A 10 , q = A 21 z 1 + A 22 z 2 + A 20 ,

where A 11 , A 12 , A 21 , A 22 , A 10 , and A 20 are stated as in Section 2.

Substituting p and q in equation (4.5), we have

(4.19) ( a 1 b 2 − a 2 b 1 ) u z 1 + ( a 1 c 2 − a 2 c 1 ) u z 2 = a 1 e q − a 2 e p .

The characteriztic equations of equation (4.19) are

d z 1 d t = a 1 b 2 − a 2 b 1 , d z 2 d t = a 1 c 2 − a 2 c 1 , d u d t = a 1 e q − a 2 e p .

Using the initial conditions: z 1 = 0 , z 2 = s , and u = u ( 0 , s ) = φ ( s ) with a parameter s , we obtain the following parametric representation for the solutions of the characteriztic equations: z 1 = ( a 1 b 2 − a 2 b 1 ) t , z 2 = ( a 1 c 2 − a 2 c 1 ) t + s , and

(4.20) u ( t , s ) = ∫ 0 t [ a 1 e q ( t , s ) − a 2 e p ( t , s ) ] d t + φ ( s ) ,

where φ ( s ) is a finite-order entire function in s ,

(4.21) p ( t , s ) = − ( 2 a 1 + b 1 α 1 + c 1 α 2 ) a 2 t + A 12 s + A 10 = − a 2 μ 1 t + A 12 s + A 10 ,

and

(4.22) q ( t , s ) = − ( 2 a 2 + b 2 α 1 + c 2 α 2 ) a 1 t + A 22 s + A 20 = − a 1 μ 2 t + A 22 s + A 20 .

If μ 1 μ 2 ≠ 0 , then it follows from equations (4.20)–(4.22) that

u ( t , s ) = 1 μ 1 e p ( t , s ) − 1 μ 2 e q ( t , s ) + φ ( s ) ,

i.e.,

(4.23) u ( z 1 , z 2 ) = 1 μ 1 e p ( z 1 , z 2 ) − 1 μ 2 e q ( z 1 , z 2 ) + φ ( s ) ,

where s = z 2 − a 1 c 2 − a 2 c 1 a 1 b 2 − a 2 b 1 z 1 . Substituting equation (4.23) into the first equation of equation (4.2), we have

a 1 φ ( s ) + b 1 a 1 c 2 − a 2 c 1 a 2 b 1 − a 1 b 2 + c 1 φ ′ ( s ) = 0 ,

which leads to

(4.24) φ ( s ) = η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] ,

where η 0 ∈ C .

If μ 1 = 0 and μ 2 ≠ 0 , i.e.,

(4.25) 2 a 1 + b 1 α 1 + c 1 α 2 = 0 .

Noting that based on the assumptions of Case 2, we have

(4.26) a 2 + b 2 A 11 + c 2 A 12 = 0 , a 1 + b 1 A 21 + c 1 A 22 = 0 .

In view of A 11 + A 21 = α 1 and A 12 + A 22 = α 2 , it yields from equations (4.25) and (4.26) that

(4.27) A 11 = A 21 = 1 2 α 1 , A 12 = A 22 = 1 2 α 2 .

Thus, we can deduce from equations (4.20) and (4.27) that

(4.28) u ( z 1 , z 2 ) = η 1 z 1 e 1 2 ( α 1 z 1 + α 2 z 2 ) + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] ,

where η 1 ∈ C satisfies b 1 b 2 η 1 2 = e β 0 .

If μ 1 ≠ 0 and μ 2 = 0 , similar to the above argument, we can obtain that u is of the form (4.28).

If μ 1 = 0 and μ 2 = 0 , similar to the above argument, we have

(4.29) u ( z 1 , z 2 ) = z 1 a 2 b 1 − a 1 b 2 ( a 2 e p ( z 1 , z 2 ) − a 1 e q ( z 1 , z 2 ) ) + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] .

In addition, the condition that μ 1 = 0 and μ 2 = 0 can lead to

(4.30) A 11 = A 21 = α 1 2 = a 2 c 1 − a 1 c 2 D , A 12 = A 22 = α 2 2 = a 1 b 2 − a 2 b 1 D .

By combining with equations (4.29) and (4.30), we have

(4.31) u ( z 1 , z 2 ) = ( η 1 z 1 + η 0 ) e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] ,

where η 1 satisfies b 1 b 2 η 1 2 = e β 0 . Thus, we obtain the conclusion (ii) of Theorem 2.5 from equations (4.23), (4.24), (4.28), and (4.31).

Therefore, this completes the proof of Theorem 2.5.□

5 Conclusion

In view of Theorems 2.1 and 2.5, we give the exact form of finite-order transcendental entire solutions of some product-type PDEs for every case. These results are some improvements of the previous results given by Lü [18], Cao and Xu [23], Xu [24], Xu et al. [19]. Meantime, our examples show that the existence of the solutions of these PDEs for every case is correct.

Acknowledgements

The authors are very thankful to referees for their valuable comments that improved the presentation of the article.

  1. Funding information: This work was supported by the National Natural Science Foundation of China (12161074), the Foundation of Education Department of Jiangxi, China (GJJ190876, GJJ202303, GJJ201813, and GJJ201343), the Talent Introduction Research Foundation of Suqian University (106-CK00042/028), and the Suqian Sci & Tech Program (Grant No. K202009).

  2. Author contributions: Conceptualization, Y. H. Xu and H. Y. Xu; writing-original draft preparation, Y. H. Xu, Y. F. Li, X. L. Liu and H. Y. Xu; writing-review and editing, H. Y. Xu, Y. F. Li, X. L. Liu and Y. H. Xu; funding acquisition, Y. H. Xu.

  3. Conflict of interest: The authors declare that none of the authors have any competing interests in the manuscript.

  4. Data availability statement: No data were used to support this study.

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Received: 2023-04-03
Revised: 2023-10-15
Accepted: 2023-10-24
Published Online: 2023-11-20

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  3. On H 2-solutions for a Camassa-Holm type equation
  4. Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type
  5. Control of multi-agent systems: Results, open problems, and applications
  6. Logical perspectives on the foundations of probability
  7. Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments
  8. A non-smooth Brezis-Oswald uniqueness result
  9. Luenberger compensator theory for heat-Kelvin-Voigt-damped-structure interaction models with interface/boundary feedback controls
  10. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part II)
  11. Positive solution for a nonlocal problem with strong singular nonlinearity
  12. Analysis of solutions for the fractional differential equation with Hadamard-type
  13. Hilfer proportional nonlocal fractional integro-multipoint boundary value problems
  14. A comprehensive review on fractional-order optimal control problem and its solution
  15. The θ-derivative as unifying framework of a class of derivatives
  16. Review Articles
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  18. Minimal-time problems for linear control systems on homogeneous spaces of low-dimensional solvable nonnilpotent Lie groups
  19. Regular Articles
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  21. An extension of the Hermite-Hadamard inequality for a power of a convex function
  22. Existence and multiplicity of solutions for a fourth-order differential system with instantaneous and non-instantaneous impulses
  23. Relay fusion frames in Banach spaces
  24. Refined ratio monotonicity of the coordinator polynomials of the root lattice of type Bn
  25. On the uniqueness of limit cycles for generalized Liénard systems
  26. A derivative-Hilbert operator acting on Dirichlet spaces
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  28. Solutions to a modified gauged Schrödinger equation with Choquard type nonlinearity
  29. A symbolic approach to multiple Hurwitz zeta values at non-positive integers
  30. Some results on the value distribution of differential polynomials
  31. Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture
  32. Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point
  33. Some results for a p(x)-Kirchhoff type variation-inequality problems in non-divergence form
  34. Homotopy cartesian squares in extriangulated categories
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  38. Binet's second formula, Hermite's generalization, and two related identities
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  69. Classification of self-adjoint domains of odd-order differential operators with matrix theory
  70. On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
  71. Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
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  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
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  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
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  80. Compression with wildcards: All exact or all minimal hitting sets
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  98. Notes on pseudodifferential operators commutators and Lipschitz functions
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  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
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  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
https://doi.org/10.1515/math-2023-0151
Keywords for this article
transcendental; entire solution; partial differential equation; Nevanlinna theory
Creative Commons
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Articles in the same Issue

  1. Special Issue on Future Directions of Further Developments in Mathematics
  2. What will the mathematics of tomorrow look like?
  3. On H 2-solutions for a Camassa-Holm type equation
  4. Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type
  5. Control of multi-agent systems: Results, open problems, and applications
  6. Logical perspectives on the foundations of probability
  7. Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments
  8. A non-smooth Brezis-Oswald uniqueness result
  9. Luenberger compensator theory for heat-Kelvin-Voigt-damped-structure interaction models with interface/boundary feedback controls
  10. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part II)
  11. Positive solution for a nonlocal problem with strong singular nonlinearity
  12. Analysis of solutions for the fractional differential equation with Hadamard-type
  13. Hilfer proportional nonlocal fractional integro-multipoint boundary value problems
  14. A comprehensive review on fractional-order optimal control problem and its solution
  15. The θ-derivative as unifying framework of a class of derivatives
  16. Review Articles
  17. On the use of L-functionals in regression models
  18. Minimal-time problems for linear control systems on homogeneous spaces of low-dimensional solvable nonnilpotent Lie groups
  19. Regular Articles
  20. Existence and multiplicity of solutions for a new p(x)-Kirchhoff problem with variable exponents
  21. An extension of the Hermite-Hadamard inequality for a power of a convex function
  22. Existence and multiplicity of solutions for a fourth-order differential system with instantaneous and non-instantaneous impulses
  23. Relay fusion frames in Banach spaces
  24. Refined ratio monotonicity of the coordinator polynomials of the root lattice of type Bn
  25. On the uniqueness of limit cycles for generalized Liénard systems
  26. A derivative-Hilbert operator acting on Dirichlet spaces
  27. Scheduling equal-length jobs with arbitrary sizes on uniform parallel batch machines
  28. Solutions to a modified gauged Schrödinger equation with Choquard type nonlinearity
  29. A symbolic approach to multiple Hurwitz zeta values at non-positive integers
  30. Some results on the value distribution of differential polynomials
  31. Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture
  32. Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point
  33. Some results for a p(x)-Kirchhoff type variation-inequality problems in non-divergence form
  34. Homotopy cartesian squares in extriangulated categories
  35. A unified perspective on some autocorrelation measures in different fields: A note
  36. Total Roman domination on the digraphs
  37. Well-posedness for bilevel vector equilibrium problems with variable domination structures
  38. Binet's second formula, Hermite's generalization, and two related identities
  39. Non-solid cone b-metric spaces over Banach algebras and fixed point results of contractions with vector-valued coefficients
  40. Multidimensional sampling-Kantorovich operators in BV-spaces
  41. A self-adaptive inertial extragradient method for a class of split pseudomonotone variational inequality problems
  42. Convergence properties for coordinatewise asymptotically negatively associated random vectors in Hilbert space
  43. Relating the super domination and 2-domination numbers in cactus graphs
  44. Compatibility of the method of brackets with classical integration rules
  45. On the inverse Collatz-Sinogowitz irregularity problem
  46. Positive solutions for boundary value problems of a class of second-order differential equation system
  47. Global analysis and control for a vector-borne epidemic model with multi-edge infection on complex networks
  48. Nonexistence of global solutions to Klein-Gordon equations with variable coefficients power-type nonlinearities
  49. On 2r-ideals in commutative rings with zero-divisors
  50. A comparison of some confidence intervals for a binomial proportion based on a shrinkage estimator
  51. The construction of nuclei for normal constituents of Bπ-characters
  52. Weak solution of non-Newtonian polytropic variational inequality in fresh agricultural product supply chain problem
  53. Mean square exponential stability of stochastic function differential equations in the G-framework
  54. Commutators of Hardy-Littlewood operators on p-adic function spaces with variable exponents
  55. Solitons for the coupled matrix nonlinear Schrödinger-type equations and the related Schrödinger flow
  56. The dual index and dual core generalized inverse
  57. Study on Birkhoff orthogonality and symmetry of matrix operators
  58. Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value
  59. Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds
  60. On semigroups of transformations that preserve a double direction equivalence
  61. Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type
  62. A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
  63. Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
  64. Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications
  65. General Randić indices of a graph and its line graph
  66. On functional reproducing kernels
  67. On the Waring-Goldbach problem for two squares and four cubes
  68. Singular moduli of rth Roots of modular functions
  69. Classification of self-adjoint domains of odd-order differential operators with matrix theory
  70. On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
  71. Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
  72. Remarks on hyponormal Toeplitz operators with nonharmonic symbols
  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
  75. Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
  76. Continued fractions related to a group of linear fractional transformations
  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
  79. On extremal cacti with respect to the first degree-based entropy
  80. Compression with wildcards: All exact or all minimal hitting sets
  81. Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
  82. Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
  83. Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
  84. On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
  86. On the quadratic residues and their distribution properties
  87. On average theta functions of certain quadratic forms as sums of Eisenstein series
  88. Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
  89. Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
  90. Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
  91. On some spaces via topological ideals
  92. Linear maps preserving equivalence or asymptotic equivalence on Banach space
  93. Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
  94. On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
  95. Entire solutions of two certain Fermat-type ordinary differential equations
  96. Generalized Lie n-derivations on arbitrary triangular algebras
  97. Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
  98. Notes on pseudodifferential operators commutators and Lipschitz functions
  99. On Graham partitions twisted by the Legendre symbol
  100. Strong limit of processes constructed from a renewal process
  101. Construction of analytical solutions to systems of two stochastic differential equations
  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
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