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Entire solutions for several secondorder partial differentialdifference equations of Fermat type with two complex variables
Advances in Difference Equations volume 2021, Article number: 52 (2021)
Abstract
This paper is concerned with description of the existence and the forms of entire solutions of several secondorder partial differentialdifference equations with more general forms of Fermat type. By utilizing the Nevanlinna theory of meromorphic functions in several complex variables we obtain some results on the forms of entire solutions for these equations, which are some extensions and generalizations of the previous theorems given by Xu and Cao (Mediterr. J. Math. 15:1–14, 2018; Mediterr. J. Math. 17:1–4, 2020) and Liu et al. (J. Math. Anal. Appl. 359:384–393, 2009; Electron. J. Differ. Equ. 2013:59–110, 2013; Arch. Math. 99:147–155, 2012). Moreover, by some examples we show the existence of transcendental entire solutions with finite order of such equations.
Introduction
The main purpose of this paper is investigation of the existence and the forms of transcendental entire solutions with finite order of secondorder differential difference equations
and
where \(g(z_{1},z_{2})\) is a polynomial in \(\mathbb{C}^{2}\). In general, for the Fermattype functional equation
Gross [6] discussed the existence of solutions of equation (1.1) and showed that the entire solutions are \(f = \cos a(z)\), \(g = \sin a(z)\) for \(m =n=2\), where \(a(z)\) is an entire function. Montel [7] proved that there are no nonconstant entire solutions for equation (1.1) for \(m=n>2\).
Recently, Han and Lü [8] gave a description of meromorphic solutions for the functional equation (1.1) when \(g(z)=f'(z)\), \(m=n\), and 1 is replaced by \(e^{\alpha z+\beta }\), where \(\alpha ,\beta \in \mathbb{C}\), and obtained the following results.
Thereom A
(See [8, Theorem 1.1])
The meromorphic solutions f of the differential equation
must be entire functions, and the following statements hold:

(A)
For \(n=1\), the general solutions of (1.2) are \(f(z)=\frac{e^{\alpha z+\beta }}{\alpha +1}+ae^{z}\) for \(\alpha \neq 1\) and \(f(z)=ze^{z+\beta }+ae^{z}\).

(B)
For \(n=2\), either \(\alpha =0\) and the general solutions of (1.2) are \(f(z)=e^{\frac{\beta }{2}}\sin (z+b)\), or \(f(z)=de^{\frac{\alpha z+\beta }{2}}\).

(C)
For \(n\geq 3\), the general solutions of (1.2) are \(f(z)=de^{\frac{\alpha z+\beta }{n}}\).
Here \(\alpha ,\beta ,a,b,d\in \mathbb{C}\) with \(d^{n}(1+(\frac{\alpha }{n})^{n})=1\) for \(n\geq 1\).
They also proved that all the trivial meromorphic solutions of \(f^{n}(z)+f^{n}(z+c)=e^{\alpha z+\beta }\) are the functions \(f(z) =de^{\frac{\alpha z+\beta }{n}}\) with \(d^{n}(1 + e^{\alpha c}) = 1\) for \(n \geq 1\) (see [8, p. 99]).
An equation is called differentialdifference equation (DDE) if the equation includes derivatives and shifts or differences of f (see [9]). In many previous papers [10–15], Naftalevich [11, 12] in 1995 discussed the meromorphic solutions of complex differentialdifference equations with one complex variable by using the operator theory and iteration method, but recently, many researchers have begun to discuss this kind of equations by using the difference analogues of Nevanlinna theory (see [16–19]). In particular, Liu et al. [3–5] investigated the existence of entire solutions with finite order of the Fermattype differentialdifference equations
They proved that the transcendental entire solutions with finite order of equation (1.3) must satisfy \(f(z) = \sin (z\pm Bi)\), where B is a constant, \(c =2k\pi \) or \(c =(2k+1)\pi \) with integer k, and the transcendental entire solutions with finite order of equation (1.4) must satisfy \(f(z) = 1 2 \sin (2z + Bi)\), where \(c =(2k+1)\pi \) with integer k, and B is a constant. In 2019, Liu and Gao [20] further studied the entire solutions of secondorder differential and difference equation with single complex variable and obtained the following:
Thereom B
(See [20, Theorem 2.1])
Let f be a transcendental entire solution with finite order of the complex differentialdifference equation
Then \(Q(z)=c_{1}c_{2}\) is a constant, and \(f(z)\) satisfies
with \(a,b\in \mathbb{C}\) such that \(a^{4}=1\) and \(c=\frac{\log (ia^{2})+2k\pi i}{a}\), \(k\in \mathbb{Z}\).
Now let us recall some previous results on Fermattype partial differential equations with several complex variables (including [21–25]). Khavinson [22] in 1995 pointed out that any entire solution of the partial differential equation \((\frac{\partial f}{\partial z_{1}} )^{2}+ ( \frac{\partial f}{\partial z_{2}} )^{2}=1\) in \(\mathbb{C}^{2}\) is necessarily linear. This partial differential equation in real variable case occurs in the study of characteristic surfaces and wave propagation theory, and it is the twodimensional eiconal equation, one of the main equations of geometric optics (see [26, 27]). In 2005, Li [28] discussed the partial differential equation of Fermattype
where g is a polynomial or an entire function in \(\mathbb{C}^{2}\), and obtained some results on the forms of entire solution of equation (1.5).
Thereom C
([28, Theorem 2.1])
Let g be a polynomial in \(\mathbb{C}^{2}\). Then u is an entire solution of the partial differential equation (1.5) if and only if

(i)
\(u=f(c_{1}z_{1}+c_{2}z_{2})\); or

(ii)
\(u=\phi _{1}(z_{1}+iz_{2})+\phi _{2}(z_{1}iz_{2})\),
where f is an entire function in \(\mathbb{C}\) satisfying \(f'(c_{1}z_{1}+c_{2}z_{2})=\pm e^{\frac{1}{2}g(z)}\), \(c_{1}\) and \(c_{2}\) are two constants satisfying \(c_{1}^{2}+c_{2}^{2}=1\), and \(\phi _{1}\) and \(\phi _{2}\) are entire functions in \(\mathbb{C}\) satisfying \(\phi '_{1}(z_{1}+iz_{2})\phi '_{2}(z_{1}iz_{2})=\frac{1}{4}e^{g(z)}\).
Very recently, Xu and Cao [1, 2, 29] investigated the existence of solutions for some Fermattype partial differentialdifference equations with several variables by using the difference logarithmic derivative lemma of several complex variables and obtained the following theorem (see [29–31]).
Thereom D
(See [1, Theorem 1.2])
Let \(c=(c_{1},c_{2})\) be a constant in \(\mathbb{C}^{2}\). Then any transcendental entire solution with finite order of the partial differentialdifference equation
has the form of \(f(z_{1}, z_{2}) = \sin (Az_{1}+B)\), where \(A\in \mathbb{C}\) is a constant satisfying \(Ae^{iAc_{1}}=1\), and \(B\in \mathbb{C}\) is a constant; in the particular case \(c_{1} = 0\), we have \(f(z_{1}, z_{2}) = \sin (z_{1}+B)\).
Theorems B, C, and D suggest the following questions as open problems.
Question 1.1
What will happen when the right side of those equations, 1, is replaced by a function \(e^{g}\) in Theorem D, where g is a polynomial in \(\mathbb{C}^{2}\)?
Question 1.2
What will happen when \(\frac{\partial f(z_{1},z_{2})}{\partial z_{1}}\) is replaced by \(\frac{\partial ^{2} f(z_{1},z_{2})}{\partial z_{1}^{2}}\) or \(\frac{\partial ^{2} f(z_{1},z_{2})}{\partial z_{1}\partial z_{2}}\) in Theorem D?
Results and some examples
In view of the above questions, this paper is concerned with description of entire solutions for several secondorder partial differentialdifference equations of Fermat type of more general form. The main tools used in this paper are the Nevanlinna theory and difference Nevanlinna theory with several complex variables. Our principal results generalize the previous theorems given by Xu and Cao [1] and Liu, Cao, and Cao [5]. Throughout this paper, for convenience, we assume that \(z+w=(z_{1}+w_{1},z_{2}+w_{2})\) for any \(z=(z_{1},z_{2})\), \(w=(w_{1},w_{2})\). We now state the main results of this paper.
Theorem 2.1
Let \(c = (c_{1}, c_{2})\in \mathbb{C}^{2}\) and \(c_{2}\neq 0\). If the partial differentialdifference equation
admits a transcendental entire solution \(f(z_{1},z_{2})\) of finite order, then \(g(z_{1},z_{2})\) must be a linear function of the form \(g(z_{1},z_{2})=A_{1}z_{1}+A_{2}z_{2}+B\), where \(A_{1}, A_{2}, B\in \mathbb{C}\). Further, \(f(z_{1},z_{2})\) must satisfy one of the following cases:

(i)
$$ f(z_{1},z_{2})=\frac{4(\xi ^{2}+1)}{A_{1}^{2}\xi } e^{\frac{1}{2}g(z_{1},z_{2})}, $$
with \(\xi (\neq 0)\), \(A_{1}, A_{2}, B\in \mathbb{C}\) satisfying
$$ \frac{\xi ^{2}1}{4(\xi ^{2}+1)i}A_{1}^{2}=e^{\frac{1}{2}(A_{1}c_{1}+A_{2}c_{2})}; $$ 
(ii)
$$ f(z_{1},z_{2})= \frac{A_{21}^{2}e^{L_{1}(z)+B_{1}}+A_{11}^{2}e^{L_{2}(z)+B_{2}}}{2A_{11}^{2}A_{21}^{2}}, $$
where \(L_{1}(z)=A_{11}z_{1}+A_{12}z_{2}+B_{1}\), \(L_{2}(z)=A_{21}z_{1}+A_{22}z_{2}+B_{2}\), \(A_{j1},A_{j2},B_{j}\in \mathbb{C}\) (\(j=1,2\)) satisfy
$$ L_{1}(z)\neq L_{2}(z),\qquad g(z)=L_{1}(z)+L_{2}(z)+B_{1}+B_{2}, $$and
$$ iA_{11}^{2}e^{L_{1}(c)}=iA_{21}^{2}e^{L_{2}(c)}=1. $$
The following examples show that the forms of solutions are precise to some extent.
Example 2.1
Let \(A_{1}=2\), \(A_{2}=1\), \(B=0\), and
Then \(\rho (f)=1\), and \(f(z_{1},z_{2})\) is a transcendental entire solution of equation (2.1) with \(g(z)=2z_{1}+z_{2}\), \(c_{1}=\pi i\), and \(c_{2}=2\pi i\).
Example 2.2
Let \(L_{1}(z)=iz_{1}+\frac{1}{2}z_{2}\), \(L_{2}(z)=iz_{1}\frac{5}{2}z_{2}\), \(B_{1}=B_{2}=0\), and
Then \(\rho (f)=1\), and \(f(z_{1},z_{2})\) is a transcendental entire solution of equation (2.1) with \(g(z)=\frac{3}{2}z_{2}\), \(c_{1}=\pi \), and \(c_{2}=\pi i\).
From Theorem 2.1 we easily get the following:
Corollary 2.1
Let \(c = (c_{1}, c_{2})\in \mathbb{C}^{2}\), \(c_{2}\neq 0\), and let \(g(z_{1},z_{2})\) be not a linear function of the form \(L(z)=A_{1}z_{1}+A_{2}z_{2}+B\), where \(A_{1},A_{2},B\in \mathbb{C}\). Then the partial differentialdifference equation
admits no transcendental entire solution of finite order.
The following example shows that the condition \(c_{2}\neq 0\) in Corollary 2.1 cannot be removed.
Example 2.3
Let \(f(z_{1},z_{2})=\frac{\sqrt{2}}{2}e^{z_{1}+z_{2}2\pi z_{2}^{2}}\). Then \(f(z_{1},z_{2})\) is a transcendental entire solution of finite order of equation (2.2) with \(c=(c_{1},c_{2})=(2\pi i, 0)\) and \(g(z_{1},z_{2})=2z_{1}+2z_{2}4\pi z_{2}^{2}\).
Remark 2.1
In addition, in view of Theorem 2.1, we can obtain the conclusions of Theorem 1.2 in [1] if \(\alpha =1\), \(\beta =0\), and \(g(z)=2k \pi i\), \(k\in \mathbb{Z}\), in equation (2.1).
For the difference counterpart of Theorem 2.1, we have the following:
Theorem 2.2
Let \(c = (c_{1}, c_{2})\in \mathbb{C}^{2}\), \(c_{2}\neq 0\). If the partial differentialdifference equation
admits a transcendental entire solution \(f(z_{1},z_{2})\) of finite order, then \(g(z_{1},z_{2})\) must be a linear function of the form \(g(z_{1},z_{2})=A_{1}z_{1}+A_{2}z_{2}+B\) with \(A_{1}, A_{2}, B\in \mathbb{C}\). Further, \(f(z_{1},z_{2})\) must satisfy one of the following cases:

(i)
$$ f(z_{1},z_{2})=\frac{4(\xi ^{2}+1)}{A_{1}^{2}\xi }e^{\frac{1}{2}g(z_{1},z_{2})}+z_{1}G_{1}(z_{2})+G_{2}(z_{2}), $$
where \(G_{1}(z_{2})\) is a finiteorder entire period function in \(z_{2}\) with period \(c_{2}\), \(\xi (\neq 0)\), \(A_{1}, A_{2}, B\in \mathbb{C}\) satisfying
$$ G_{2}(z_{2}+c_{2})=G_{2}(z_{2})c_{1}G_{1}(z_{2}),\qquad \frac{\xi ^{2}1}{2i(\xi ^{2}+1)} A_{1}^{2}+1=e^{\frac{1}{2}(A_{1}c_{1}+A_{2}c_{2})}=e^{ \frac{1}{2}g(c_{1},c_{2})}; $$ 
(ii)
$$ f(z_{1},z_{2})= \frac{A_{21}^{2}e^{L_{1}(z)+B_{1}}+A_{11}^{2}e^{L_{2}(z)+B_{2}}}{2 A_{11}^{2}A_{21}^{2}}+z_{1}G_{1}(z_{2})+G_{2}(z_{2}), $$
where \(G_{1}(z_{2})\) is a finiteorder entire period function in \(z_{2}\) with period \(c_{2}\), \(L_{1}(z)=A_{11}z_{1}+A_{12}z_{2}+B_{1}\), \(L_{2}(z)=A_{21}z_{1}+A_{22}z_{2}+B_{2}\), \(A_{ij},B_{i}\in \mathbb{C}\) satisfy
$$\begin{aligned}& G_{2}(z_{2}+c_{2})=G_{2}(z_{2})c_{1}G_{1}(z_{2}),\qquad L_{1}(z)\neq L_{2}(z), \\& g(z)=L_{1}(z)+L_{2}(z)+B_{1}+B_{2}, \end{aligned}$$and
$$ \bigl(1i A_{11}^{2}\bigr)e^{(A_{11}c_{1}+A_{12}c_{2})}=1,\qquad \bigl(1+i A_{21}^{2}\bigr)e^{(A_{21}c_{1}+A_{22}c_{2})}=1. $$
The following examples explain the existence of transcendental finiteorder entire solutions of (2.3).
Example 2.4
Let \(A_{1}=2\), \(A_{2}=1\), \(G_{1}(z_{2})=e^{z_{2}}\), \(G_{2}(z_{2})=e^{2z_{2}}z_{2}e^{z_{2}}\), \(B=0\), and
Then \(\rho (f)=1\), and \(f(z_{1},z_{2})\) is a transcendental entire solution of equation (2.3) with \(g(z)=2z_{1}z_{2}\), \(c_{1}=2\pi i\), and \(c_{2}=2\pi i\).
Example 2.5
Let \(L_{1}(z)=z_{1}+z_{2}\), \(L_{2}(z)=z_{1}z_{2}\), \(G_{1}(z)=e^{\frac{4\pi i}{\log (i)}z_{2}}\), \(G_{2}(z_{2})=\frac{\log (i)}{\log 2}e^{ \frac{4\pi i z_{2}}{\log (i)}}\), \(B_{1}=B_{2}=0\), and
Then \(f(z_{1},z_{2})\) is a transcendental finiteorder entire solution of equation (2.3) with \(g(z)=2z_{1}\), \(c_{1}=\frac{1}{2}\log 2\), and \(c_{2}=\frac{1}{2}\log (i)\).
In view of Theorem 2.2, we obtain the following:
Corollary 2.2
Let \(c = (c_{1}, c_{2})\in \mathbb{C}^{2}\), \(c_{2}\neq 0\), and let \(g(z_{1},z_{2})\) be not a linear function of the form \(L(z)=A_{1}z_{1}+A_{2}z_{2}+B\) with \(A_{1},A_{2},B\in \mathbb{C}\). Then the partial differentialdifference equation
has no transcendental entire solution of finite order.
The following example shows that the condition \(c_{2}\neq 0\) in Corollary 2.2 cannot be removed.
Example 2.6
Let \(f(z_{1},z_{2})=e^{z_{1}+z_{2}4\pi i z_{2}^{3}}\). Then \(f(z_{1},z_{2})\) is a transcendental finiteorder entire solution of equation (2.4) with \(c=(c_{1},c_{2})=(2\pi i, 0)\) and \(g(z_{1},z_{2})=2z_{1}+2z_{2}8\pi i z_{2}^{3}\).
When \(\frac{\partial ^{2} f(z_{1},z_{2})}{\partial z_{1}^{2}}\) is replaced by \(\frac{\partial ^{2} f(z_{1},z_{2})}{\partial z_{1}\partial z_{2}}\) in Theorems 2.1 and 2.2, we have the following:
Theorem 2.3
Let \(c = (c_{1}, c_{2})\in \mathbb{C}^{2}\), \(c_{1}\neq 0\), \(c_{2}\neq 0\). If the partial differentialdifference equation
admits a transcendental entire solution of finite order, then \(g(z_{1},z_{2})\) must be a linear function of the form \(g(z_{1},z_{2})=A_{1}z_{1}+A_{2}z_{2}+B\) with \(A_{1}, A_{2}, B\in \mathbb{C}\). Further, \(f(z_{1},z_{2})\) must satisfy one of the following cases:

(i)
$$ f(z_{1},z_{2})=\frac{4(\xi ^{2}+1)}{A_{1} A_{2}\xi } e^{\frac{1}{2}g(z_{1},z_{2})} $$
with \(\xi (\neq 0)\), \(A_{1}, A_{2}, B\in \mathbb{C}\) satisfying
$$ \frac{\xi ^{2}1}{4(\xi ^{2}+1)i}A_{1} A_{2}=e^{\frac{1}{2}(A_{1}c_{1}+A_{2}c_{2})}; $$ 
(ii)
$$ f(z_{1},z_{2})= \frac{A_{21}A_{22}e^{L_{1}(z)+B_{1}}+A_{11}A_{12}e^{L_{2}(z)+B_{2}}}{2A_{11}A_{12}A_{21}A_{22}}, $$
where \(L_{1}(z)=A_{11}z_{1}+A_{12}z_{2}+B_{1}\), \(L_{2}(z)=A_{21}z_{1}+A_{22}z_{2}+B_{2}\), \(A_{j1},A_{j2},B_{j}\in \mathbb{C}\) (\(j=1,2\)) satisfy
$$ L_{1}(z)\neq L_{2}(z),\qquad g(z)=L_{1}(z)+L_{2}(z)+B_{1}+B_{2}, $$and
$$ iA_{11}A_{12}e^{L_{1}(c)}=iA_{21}A_{22}e^{L_{2}(c)}=1. $$
Example 2.7
Let \(A_{1}=2\), \(A_{2}=2\), \(B=0\), and
Then \(\rho (f)=1\), and \(f(z_{1},z_{2})\) is a transcendental entire solution of equation (2.5) with \(g(z)=2z_{1}+2z_{2}\), \(c_{1}=\pi i\), and \(c_{2}=\pi i\).
Example 2.8
Let \(L_{1}(z)=z_{1}+z_{2}\), \(L_{2}(z)=z_{1}z_{2}\), \(B_{1}=B_{2}=0\), and
Then \(\rho (f)=1\), and \(f(z_{1},z_{2})\) is a transcendental entire solution of equation (2.5) with \(g(z)=2z_{1}\), \(c_{1}=\frac{\pi }{2}i\), and \(c_{2}=\pi i\).
From Theorem 2.3 we get the following:
Corollary 2.3
Let \(c = (c_{1}, c_{2})\in \mathbb{C}^{2}\), \(c_{1}\neq 0\), \(c_{2}\neq 0\), and let \(g(z_{1},z_{2})\) be not a linear function of the form \(L(z)=A_{1}z_{1}+A_{2}z_{2}+B\) with \(A_{1},A_{2},B\in \mathbb{C}\). Then the partial differentialdifference equation
admits no transcendental entire solution of finite order.
The following example shows that the condition \(c_{1}\neq 0\), \(c_{2}\neq 0\) in Corollary 2.3 cannot be removed.
Example 2.9
Let \(f(z_{1},z_{2})=e^{z_{2}+ z_{2}^{3}}\). Then \(f(z_{1},z_{2})\) is a transcendental finiteorder entire solution of equation (2.6) with \(c=(c_{1},c_{2})=(2\pi i, 0)\) and \(g(z_{1},z_{2})=2z_{2}+2z_{2}^{3}\).
Theorem 2.4
Let \(c = (c_{1}, c_{2})\neq (0,0)\in \mathbb{C}^{2}\). If the partial differentialdifference equation
admits a transcendental entire solution of finite order, then \(g(z_{1},z_{2})\) must be a linear function of the form \(g(z_{1},z_{2})=A_{1}z_{1}+A_{2}z_{2}+B\) with \(A_{1}, A_{2}, B\in \mathbb{C}\). Further, \(f(z_{1},z_{2})\) must satisfy one of the following cases:

(i)
$$ f(z_{1},z_{2})=\frac{4(\xi ^{2}+1)}{A_{1}A_{2}\xi }e^{\frac{1}{2}g(z_{1},z_{2})}+G_{3}(z_{1})+D_{1}z_{1}+G_{4}(z_{2})+D_{2}z_{2}, $$
where \(G_{3}(z_{1})\) and \(G_{4}(z_{2})\) are finiteorder entire periodic functions in \(z_{1}\) and \(z_{2}\) with periods \(c_{1}\) and \(c_{2}\), respectively, and \(\xi (\neq 0)\), \(A_{1}, A_{2}, B,D_{1},D_{2}\in \mathbb{C}\) satisfy
$$ \frac{\xi ^{2}1}{4i(\xi ^{2}+1)} A_{1}A_{2}+1=e^{\frac{1}{2}(A_{1}c_{1}+A_{2}c_{2})}=e^{ \frac{1}{2}g(c_{1},c_{2})},\qquad D_{1}c_{1}+D_{2}c_{2}=0; $$ 
(ii)
$$ f(z_{1},z_{2})= \frac{A_{21}A_{22}e^{L_{1}(z)+B_{1}}+A_{11}A_{12}e^{L_{2}(z)+B_{2}}}{2A_{11}A_{12}A_{21}A_{22}}+G_{3}(z_{1})+D_{1}z_{1}+G_{4}(z_{2})+D_{2}z_{2}, $$
where \(G_{3}(z_{1})\) and \(G_{4}(z_{2})\) are finiteorder entire periodic functions in \(z_{1}\) and \(z_{2}\) with periods \(c_{1}\) and \(c_{2}\), respectively, \(L_{1}(z)=A_{11}z_{1}+A_{12}z_{2}+B_{1}\), \(L_{2}(z)=A_{21}z_{1}+A_{22}z_{2}+B_{2}\), \(A_{j1}, A_{j2}, B_{j}\in \mathbb{C}\) (\(j=1,2\)) satisfy
$$ L_{1}(z)\neq L_{2}(z),\qquad g(z)=L_{1}(z)+L_{2}(z)+B_{1}+B_{2},\qquad D_{1}c_{1}+D_{2}c_{2}=0, $$and
$$ (1i A_{11}A_{12})e^{(A_{11}c_{1}+A_{12}c_{2})}=1,\qquad (1+i A_{21}A_{22})e^{(A_{21}c_{1}+A_{22}c_{2})}=1. $$
Example 2.10
Let \(A_{1}=2\), \(A_{2}=2\), \(B=0\), \(G_{3}(z_{1})=e^{2z_{1}}\), \(G_{4}(z_{2})=e^{4z_{2}}\), and
Then \(\rho (f)=1\), and \(f(z_{1},z_{2})\) is a transcendental entire solution of equation (2.7) with \(g(z)=2z_{1}+2z_{2}\), \(c_{1}=\pi i\), and \(c_{2}=\frac{\pi }{2} i\).
Example 2.11
Let \(L_{1}(z)=z_{1}+z_{2}\), \(L_{2}(z)=z_{1}2z_{2}\), \(B_{1}=B_{2}=0\), \(G_{3}(z)=e^{\frac{6\pi i}{\log [2(2+i)]}z_{1}}\), \(G_{4}(z)=e^{\frac{6\pi i}{\log (1i)\log (12i)]}z_{2}}\), and
Then \(\rho (f)=1\), and \(f(z_{1},z_{2})\) is a transcendental entire solution of equation (2.7) with \(g(z)=2z_{1}z_{2}\), \(c_{1}=\frac{\log [2(2+i)]}{3}\), and \(c_{2}=\frac{\log (1i)\log (12i)}{3}\).
In view of Theorem 2.4, we obtain the following:
Corollary 2.4
Let \(c = (c_{1}, c_{2})\neq (0,0) \in \mathbb{C}^{2}\), and let \(g(z_{1},z_{2})\) be not a linear function of the form \(L(z)=A_{1}z_{1}+A_{2}z_{2}+B\) with \(A_{1},A_{2},B\in \mathbb{C}\). Then the partial differentialdifference equation
admits no transcendental entire solution of finite order.
In view of Theorems 2.1 and 2.3, we also get the following:
Corollary 2.5
Let f be a finiteorder transcendental entire solution of the partial differential equation
Then \(f(z_{1},z_{2})\) must be of the form
where \(L(z)=A_{1}z_{1}+A_{2}z_{2}\) with \(A_{1},A_{2},B\in \mathbb{C}\) satisfying \(A_{1}^{4}=1\) and \(A_{1}^{2}A_{2}^{2}=1\).
Some lemmas
The following lemmas play the key role in proving our results.
Lemma 3.1
For an entire function F on \(\mathbb{C}^{n}\) with \(F(0)\neq 0\), put \(\rho (n_{F}) =\rho <\infty \). Then there exist a canonical function \(f_{F}\) and a function \(g_{F}\in \mathbb{C}^{n}\) such that \(F(z) = f_{F} (z)e^{g_{F} (z)}\). For the particular case \(n = 1\), \(f_{F}\) is the canonical Weierstrass product.
Remark 3.1
Here \(\rho (n_{F} )\) is the order of the counting function of zeros of F.
Lemma 3.2
([34])
If g and h are entire functions on the complex plane \(\mathbb{C}\) and \(g(h)\) is an entire function of finite order, then there are only two possible cases:

(a)
the internal function h is a polynomial, and the external function g is of finite order;

(b)
the internal function h is not a polynomial but a function of finite order, and the external function g is of zero order.
Lemma 3.3
([35, Theorem 1.106])
Suppose that \(a_{0}(z),a_{1}(z),\ldots ,a_{n}(z)\) (\(n\geq 1\)) are meromorphic functions on \(\mathbb{C}^{m}\) and \(g_{0}(z),g_{1}(z),\ldots ,g_{n}(z)\) are entire functions on \(\mathbb{C}^{m}\) such that \(g_{j}(z)g_{k}(z)\) are not constants for \(0\leq j< k\leq n\). If
and
where \(T(r)=\min_{0\leq j< k\leq n}T(r,e^{g_{j}g_{k}})\), then \(a_{j}(z)\equiv 0\) (\(j=0,1,2,\ldots ,n\)).
Lemma 3.4
([35, Lemma 3.1])
Let \(f_{j}(\not \equiv 0)\), \(j = 1,2,3\), be meromorphic functions on \(\mathbb{C}^{m}\) such that \(f_{1}\) is not constant, \(f_{1}+f_{2}+f_{3}=1\), and
for all r outside possibly a set of finite logarithmic measure, where \(\lambda <1\) is a positive number. Then either \(f_{2}=1\) or \(f_{3}=1\).
Remark 3.2
Here \(N_{2}(r, \frac{1}{f})\) is the counting function of zeros of f in \(z\leq r\), where the simple zero is counted once, and the multiple zero is counted twice.
The proof of Theorem 2.1
Proof
Let \(f(z_{1},z_{2})\) be a transcendental finiteorder entire solution of equation (2.1). We first rewrite (2.1) in the form
or
Since f is a finiteorder transcendental entire function and g is a polynomial, by Lemmas 3.1 and 3.2 there exists a polynomial \(p(z)\) such that
Denote
By combining with (4.2) it follows that
This leads to
where
We consider two cases.
Case 1. If \(e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}\) is a constant, then \(\gamma _{2}(z+c)\gamma _{1}(z+c)\) is a constant. Set \(\gamma _{2}(z+c)\gamma _{1}(z+c)=\kappa \), \(\kappa \in \mathbb{C}\). In view of (4.3), \(p(z)\) is a constant. Let \(\xi =e^{p(z)}\). Then equations (4.4)–(4.5) can be represented as
where \(K_{1}=\frac{\xi +\xi ^{1}}{2}\), \(K_{2}=\frac{\xi \xi ^{1}}{2i}\), and \(K_{1}^{2}+K_{2}^{2}=1\).
This leads to
Since \(g(z)\) is a polynomial, (4.8) implies that \(g(z+c)g(z)\) is a constant in \(\mathbb{C}\). Otherwise, we obtain a contradiction from the fact that the lefthand side of this equation is not transcendental but the righthand side is transcendental. Thus it follows that \(g(z)=L(z)+H(s)+B\), where \(L(z)=A_{1}z_{1}+A_{2}z_{2}\), \(A_{1}\neq 0\), and \(H(s)\) is a polynomial in s in \(\mathbb{C}\), \(s=c_{2}z_{1}c_{1}z_{2}\).
We will prove that \(H(s)\equiv 0\). If \(\deg _{s}H=n\), then equation (4.8) implies
that is,
where \(\zeta _{0}\in \mathbb{C}\). By comparing the degree of s in both sides of the above equation we have \(2(n1)=n1\), that is, \(n=1\). Thus the form of \(L(z)+H(s)+B\) is still the linear form of \(A_{1}z_{1}+A_{2}z_{2}+B\), which means that \(H(s)\equiv 0\). Hence it follows that \(g(z)=L(z)+B=A_{1}z_{1}+A_{2}z_{2}+B\). By combining with (4.6)–(4.8) we conclude that
which implies that
This completes the proof of Theorem 2.1(i).
Case 2. \(e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}\) is not a constant. Obviously, \(Q_{1}(z)\equiv 0\) and \(Q_{2}(z)\equiv 0\) cannot hold at the same time. Otherwise, it would follows from (4.6) that \(e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}=1\), a contradiction. If \(Q_{1}(z)\equiv 0\) and \(Q_{2}(z)\not \equiv 0\), then from (4.6) this yields that
Thus we conclude that \(e^{\gamma _{2}(z)\gamma _{1}(z+c)}\) is a nonconstant because \(e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}\) is not a constant. Moreover, it follows that \(e^{\gamma _{2}(z+c)\gamma _{2}(z)}\) is not a constant. Otherwise, \(\gamma _{2}(z+c)=\gamma _{2}(z)+\zeta \), where \(\zeta \in \mathbb{C}\). Then from (4.10) we have \([iQ_{2}(z)e^{\zeta }1]e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}\equiv 1\), which is a contradiction with the nonconstant \(e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}\). Thus (4.10) can be written in the form
By applying Lemma 3.3 for (4.11) we easily get a contradiction. If \(Q_{2}(z)\equiv 0\) and \(Q_{1}(z)\not \equiv 0\), by using the same argument as before, we can get a contradiction. Hence we have that \(Q_{1}(z)\not \equiv 0\) and \(Q_{2}(z)\not \equiv 0\).
Since \(\gamma _{1}(z)\), \(\gamma _{2}(z)\) are polynomials and \(e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}\) is not a constant, by applying Lemma 3.4 to (4.6) it follows that
Subcase 2.1. Suppose that \(iQ_{1}(z)e^{\gamma _{1}(z)\gamma _{1}(z+c)}\equiv 1\). Then it follows from (4.6) that \(iQ_{2}(z)e^{\gamma _{2}(z)\gamma _{2}(z+c)}\equiv 1\). This means that \(\gamma _{1}(z)\gamma _{1}(z+c)=\zeta _{1}\) and \(\gamma _{2}(z)\gamma _{2}(z+c)=\zeta _{2}\), where \(\zeta _{1},\zeta _{2}\in \mathbb{C}\). Hence we have that \(\gamma _{1}(z)=L_{1}(z)+H_{1}(s)+B_{1}\) and \(\gamma _{2}(z)=L_{2}(z)+H_{2}(s)+B_{2}\), where \(L_{j}(z)=A_{j1}z_{1}+A_{j2}z_{2}\), \(H_{j}(s)\), \(j=1,2\), are polynomials in \(s=c_{2}z_{1}c_{1}z_{2}\), \(A_{j1}, A_{j2}, B_{j} \in \mathbb{C}\), \(j=1,2\). In view of the definitions of \(Q_{1}\), \(Q_{2}\), similarly to the argument in Case 1, we can conclude that \(H_{1}(s)=H_{2}(s)\equiv 0\). In addition, it follows that \(L_{1}(z)\neq L_{2}(z)\). Otherwise, \(\gamma _{2}(z+c)\gamma _{1}(z+c)\) is a constant, which implies that \(e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}\) is a constant, a contradiction. Substituting these into (4.12), we have
By combining with (4.5) we have
From the definitions of \(\gamma _{1}(z)\) and \(\gamma _{2}(z)\) we can see that
where \(L(z)=L_{1}(z)+L_{2}(z)\), \(B=B_{1}+B_{2}\).
Subcase 2.2. Suppose that \(iQ_{2}(z)e^{\gamma _{2}(z)\gamma _{1}(z+c)}\equiv 1\). Then it follows from (4.6) that \(iQ_{1}(z)e^{\gamma _{1}(z)\gamma _{2}(z+c)}\equiv 1\). This means that \(\gamma _{2}(z)\gamma _{1}(z+c)=\zeta _{1}\) and \(\gamma _{1}(z)\gamma _{2}(z+c)=\zeta _{2}\), where \(\zeta _{1},\zeta _{2}\in \mathbb{C}\). Thus it follows that \(\gamma _{1}(z+2c)\gamma _{1}(z)=\zeta _{1}\zeta _{2}\) and \(\gamma _{2}(z+c)\gamma _{2}(z)=\zeta _{1}\zeta _{2}\). We can obtain that \(\gamma _{1}(z)=L(z)+H(s)+B_{1}\) and \(\gamma _{2}(z)=L(z)+H(s)+B_{2}\), where \(L(z)=a_{1}z_{1}+a_{2}z_{2}\), and \(H(s)\) is a polynomial in \(s=c_{2}z_{1}c_{1}z_{2}\), \(a_{1},a_{2}, B_{1},B_{2}\in \mathbb{C}\). This yields that \(\gamma _{2}(z+c)\gamma _{1}(z+c)=B_{2}B_{1}\), which implies that \(e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}\) is a constant, a contradiction.
This completes the proof of Theorem 2.1. □
The proof of Theorem 2.2
Proof
Let \(f(z_{1},z_{2})\) be a finiteorder transcendental entire solution of equation (2.3). We first rewrite (2.3) in the form
or
Since f is a finiteorder transcendental entire function and g is a polynomial, by Lemmas 3.1 and 3.2 there exists a polynomial \(p(z)\) in \(\mathbb{C}^{2}\) such that
Denote
By combining with (5.2) it follows that
This leads to
where
We consider two cases.
Case 1. If \(e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}\) is a constant, then \(\gamma _{2}(z+c)\gamma _{1}(z+c)\) is a constant. Set \(\gamma _{2}(z+c)\gamma _{1}(z+c)=\kappa \), \(\kappa \in \mathbb{C}\). In view of (5.3), this yields that \(p(z)\) is a constant. Let \(\xi =e^{p(z)}\). Then equations (5.4)–(5.5) can be represented as
where \(K_{1}=\frac{\xi +\xi ^{1}}{2}\), \(K_{2}=\frac{\xi \xi ^{1}}{2i}\), and \(K_{1}^{2}+K_{2}^{2}=1\).
This leads to
Since \(g(z)\) is a polynomial, (5.8) implies \(g(z+c)g(z)\), and thus \(e^{\frac{g(z+c)g(z)}{2}}\) must be a constant. Denote \(g(z+c)g(z)=\zeta \), where ζ is a constant in \(\mathbb{C}\). By using the same argument as in Case 1 of Theorem 2.1, we obtain that \(g(z)=L(z)+B\), where \(L(z)=A_{1}z_{1}+A_{2}z_{2}\), \(B\in \mathbb{C}\).
By combining with (5.8) it follows that
Solving the first equation in (5.7), we have
Substituting (5.10) into the second equation in (5.7) and combining with (5.9), we get that \(G_{1}(z_{2}+c_{2})=G_{1}(z_{2})\) and \(G_{2}(z_{2}+c_{2})G_{2}(z_{2})=c_{1}G_{1}(z_{2})\), which means that \(G_{1}(z_{2})\) is a finiteorder entire period functions in \(z_{2}\) with period \(c_{2}\).
Case 2. \(e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}\) is not a constant. Obviously, \(Q_{3}(z)\equiv 0\) and \(Q_{4}(z)\equiv 0\) cannot hold at the same time. Otherwise, it would follow from (5.6) that \(e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}=1\), a contradiction. If \(Q_{3}(z)\equiv 0\) and \(Q_{4}(z)\not \equiv 0\), then from (5.6) it follows that
Thus we conclude that \(e^{\gamma _{2}(z)\gamma _{1}(z+c)}\) is not a constant because \(e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}\) is not a constant. Moreover, it follows that \(e^{\gamma _{2}(z+c)\gamma _{2}(z)}\) is not a constant. Otherwise, \(\gamma _{2}(z+c)=\gamma _{2}(z)+\zeta \), where \(\zeta \in \mathbb{C}\). Then from (5.11) we have \([Q_{4}(z)e^{\zeta }1]e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}\equiv 1\), which is a contradiction with the nonconstant \(e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}\). Thus (5.11) can be written in the form
By applying Lemma 3.3 to (5.12) we easily get a contradiction. If \(Q_{4}(z)\equiv 0\) and \(Q_{3}(z)\not \equiv 0\), by using the same argument as before we can get a contradiction. Hence we have that \(Q_{3}(z)\not \equiv 0\) and \(Q_{4}(z)\not \equiv 0\).
Since \(\gamma _{1}(z)\), \(\gamma _{2}(z)\) are polynomials and \(e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}\) is a nonconstant, by applying Lemma 3.4 to (5.6) it follows that
Subcase 2.1. Suppose that \(Q_{3}(z)e^{\gamma _{1}(z)\gamma _{1}(z+c)}\equiv 1\). Then it follows from (5.6) that \(Q_{4}(z)e^{\gamma _{2}(z)\gamma _{2}(z+c)}\equiv 1\). This means that \(\gamma _{1}(z)\gamma _{1}(z+c)=\zeta _{1}\), \(\gamma _{2}(z)\gamma _{2}(z+c)= \zeta _{2}\), where \(\zeta _{1},\zeta _{2}\in \mathbb{C}\). Hence we have that \(\gamma _{1}(z)=L_{1}(z)+H_{1}(s)+B_{1}\) and \(\gamma _{2}(z)=L_{2}(z)+H_{2}(s)+B_{2}\), where \(L_{j}(z)=A_{j1}z_{1}+A_{j2}z_{2}\), \(H_{j}(s_{1})\), \(j=1,2\), are polynomials in \(s_{1}=c_{2}z_{1}c_{1}z_{2}\), \(A_{j1}, A_{j2}, B_{j} \in \mathbb{C}\), \(j=1,2\). Similarly to the argument in Case 1 of Theorem 2.2, we have \(H_{1}(s)=H_{2}(s)\equiv 0\). Thus it follows that \(\gamma _{1}(z)=L_{1}(z)+B_{1}\) and \(\gamma _{2}(z)=L_{2}(z)+B_{2}\). Obviously, \(L_{1}(z)\neq L_{2}(z)\). Otherwise, \(\gamma _{2}(z+c)\gamma _{1}(z+c)\) is a constant, which implies that \(e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}\) is a constant, a contradiction. Substituting these into (5.6), we have
By solving the equation
we have
Substituting (5.16) into (5.5) and combining with (5.14), we have \(G_{1}(z_{2}+c_{2})=G_{1}(z_{2})\) and \(G_{2}(z_{2}+c_{2})G_{2}(z_{2})=c_{1}G_{1}(z_{2})\), which means that \(G_{1}(z_{2})\) is a finiteorder entire periodic function in \(z_{2}\) with period \(c_{2}\).
From the definitions of \(\gamma _{1}(z)\) and \(\gamma _{2}(z)\) we can see that
where \(L(z)=L_{1}(z)+L_{2}(z)\), \(B=B_{1}+B_{2}\).
Subcase 2.2. Suppose that \(Q_{4}(z)e^{\gamma _{2}(z)\gamma _{1}(z+c)}\equiv 1\). Similarly to the argument in Subcase 2.2 in Theorem 2.1, we can get a contradiction.
Therefore this completes the proof of Theorem 2.2. □
Proofs of Theorems 2.3 and 2.4
Proof of Theorem 2.4
Suppose that \(f(z_{1},z_{2})\) is a finiteorder transcendental entire solution of equation (2.7). We first rewrite (2.7) in the form
Since f is a finiteorder transcendental entire function and g is a polynomial, by Lemmas 3.1 and 3.2 there exists a polynomial \(p(z)\) such that
Denote
By combining with (6.2) it follows that
This leads to
where
We consider two cases.
Case 1. If \(e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}\) is a constant, then \(\gamma _{2}(z+c)\gamma _{1}(z+c)\) is a constant. Set \(\gamma _{2}(z+c)\gamma _{1}(z+c)=\kappa \), \(\kappa \in \mathbb{C}\). In view of (6.3), this yields that \(p(z)\) is a constant. Let \(\xi =e^{p(z)}\). Then equations (6.4)–(6.5) become
where \(K_{1}=\frac{\xi +\xi ^{1}}{2}\), \(K_{2}=\frac{\xi \xi ^{1}}{2i}\), and \(K_{1}^{2}+K_{2}^{2}=1\).
This leads to
Since \(g(z)\) is a polynomial, (6.8) implies \(g(z+c)g(z)\), and thus \(e^{\frac{g(z+c)g(z)}{2}}\) must be a constant. Denote \(g(z+c)g(z)=\zeta \), where ζ is a constant in \(\mathbb{C}\). Thus it follows that \(g(z)=L(z)+H(s)+B\), where \(L(z)=A_{1}z_{1}+A_{2}z_{2}\), and \(H(s)\) is a polynomial in s in \(\mathbb{C}\), \(s=c_{2}z_{1}c_{1}z_{2}\). Substituting this into (6.8), we deduce that
Since \(g(z)\) is a polynomial, then (6.9) implies that \(g(z+c)g(z)\) is a constant in \(\mathbb{C}\). Otherwise, we would obtain a contradiction from the fact that the lefthand side of the above equation is not transcendental but the righthand side is transcendental. Hence it follows that
where \(\zeta _{0}\in \mathbb{C}\). If \(c_{1}=0\), \(c_{2}\neq 0\), that is, \(\frac{K_{2}}{2K_{1}} (\frac{1}{2}A_{2}c_{2}H'+\frac{1}{2}A_{1}A_{2} )+1=\zeta _{0}\). Thus, either \(A_{2}=0\), or \(H'\) is a constant. If \(A_{2}=0\), then \(\zeta _{0}=1\), that is, \(e^{\frac{g(z+c)g(z)}{2}}\) is a constant. By combining with \(c_{1}=0\) this means that \(g(z)\) is a constant. Set \(e^{\frac{g}{2}}=\theta \). In view of the first equation of (6.5), we have
where \(\mu (z_{1})\) is a finiteorder transcendental entire function. Substituting this into the second equation of (6.5), we have
Combining with \(c_{1}=0\), this yields that \(K_{1}\theta c_{2} z_{1}=K_{2} \theta \), which is impossible. Hence \(H'\) is a constant, that is, \(H(s)=c2z_{1}\).
If \(c_{2}=0\), \(c_{1}\neq 0\), similarly to the above argument, we can obtain that \(H(s)=c_{1}z_{2}\).
Let \(c_{1}\neq 0\) and \(c_{2}\neq 0\). If \(A_{2}c_{2}A_{1}c_{1}=0\), noting that the lefthand side of (6.10) is a constant, we have \(\deg _{s} H\leq 1\), that is, \(H(s)=c_{2}z_{1}c_{1}z_{2}+\tau \), where \(\tau \in \mathbb{C}\). If \(A_{2}c_{2}A_{1}c_{1}\neq 0\), we easily obtain that \(\deg _{s} H\leq 1\), that is, \(H(s)=c_{2}z_{1}c_{1}z_{2}++\tau \), where \(\tau \in \mathbb{C}\). Thus the form of \(L(z)+H(s)+B\) is still the linear form of \(A_{1}z_{1}+A_{2}z_{2}+B\), which means that \(H(s)\equiv 0\). Hence we obtain that \(g(z)=L(z)+B\), where \(L(z)=A_{1}z_{1}+A_{2}z_{2}\), \(B\in \mathbb{C}\).
By combining with (6.8) it follows that
Solving the first equation in (6.7), we have
Substituting (6.13) into the second equation in (6.7) and combining with (6.12), we get that
which yields that \(\phi (z_{1})=G_{3}(z_{1})+D_{1}z_{1}\) and \(\varphi (z_{2})=G_{4}(z_{2})+D_{2}z_{2}\), where \(D_{1}c_{1}+D_{2}c_{2}=0\) and \(G_{3}(z_{1})\), \(G_{4}(z_{2})\) are finiteorder entire period functions in \(z_{1}\), \(z_{2}\) with periods \(c_{1}\), \(c_{2}\), respectively.
Case 2. \(e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}\) is not a constant. Obviously, \(Q_{5}(z)\equiv 0\) and \(Q_{6}(z)\equiv 0\) cannot hold at the same time. Otherwise, it would follow from (6.6) that \(e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}=1\), a contradiction. If \(Q_{5}(z)\equiv 0\) and \(Q_{6}(z)\not \equiv 0\), then from (6.6) we get that
Thus we conclude that \(e^{\gamma _{2}(z)\gamma _{1}(z+c)}\) is not a constant because \(e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}\) is not a constant. Moreover, it follows that \(e^{\gamma _{2}(z+c)\gamma _{2}(z)}\) is not a constant. Otherwise, \(\gamma _{2}(z+c)=\gamma _{2}(z)+\zeta \), where \(\zeta \in \mathbb{C}\). Then from (6.14) we have \([Q_{6}(z)e^{\zeta }1]e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}\equiv 1\), which is a contradiction with the nonconstant \(e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}\). Thus (6.14) can be written in the form
By applying Lemma 3.3 for (6.15) we easily get a contradiction. If \(Q_{6}(z)\equiv 0\) and \(Q_{5}(z)\not \equiv 0\), by using the same argument as before we can get a contradiction. Hence we have that \(Q_{5}(z)\not \equiv 0\) and \(Q_{6}(z)\not \equiv 0\).
Since \(\gamma _{1}(z)\), \(\gamma _{2}(z)\) are polynomials and \(e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}\) is not a constant, by applying Lemma 3.4 to (6.6) it follows that
Subcase 2.1. Suppose that \(Q_{5}(z)e^{\gamma _{1}(z)\gamma _{1}(z+c)}\equiv 1\). Then it follows from (6.6) that \(Q_{6}(z)e^{\gamma _{2}(z)\gamma _{2}(z+c)}\equiv 1\). This means that \(\gamma _{1}(z)\gamma _{1}(z+c)=\zeta _{1}\), \(\gamma _{2}(z)\gamma _{2}(z+c)= \zeta _{2}\), where \(\zeta _{1},\zeta _{2}\in \mathbb{C}\). Hence we have that \(\gamma _{1}(z)=L_{1}(z)+H_{1}(s)+B_{1}\) and \(\gamma _{2}(z)=L_{2}(z)+H_{2}(s)+B_{2}\), where \(L_{j}(z)=A_{j1}z_{1}+A_{j2}z_{2}\), \(H_{j}(s)\), \(j=1,2\), are polynomials in \(s=c_{2}z_{1}c_{1}z_{2}\), \(A_{j1}, A_{j2}, B_{j} \in \mathbb{C}\), \(j=1,2\). Similarly to the argument in Case 1, we have \(H_{1}(s)=H_{2}(s)\equiv 0\). Thus it follows that \(\gamma _{1}(z)=L_{1}(z)+B_{1}\) and \(\gamma _{2}(z)=L_{2}(z)+B_{2}\). Obviously, \(L_{1}(z)\neq L_{2}(z)\). Otherwise, \(\gamma _{2}(z+c)\gamma _{1}(z+c)\) would be a constant, which implies that \(e^{\gamma _{2}(z+c)\gamma _{1}(z+c)}\) is a constant, a contradiction. Substituting these into (6.6), we have
By solving the equation
we have
Substituting (6.19) into (6.5) and combining with (6.17), we get that
which yields that \(\phi (z_{1})=G_{3}(z_{1})+D_{1}z_{1}\) and \(\varphi (z_{2})=G_{4}(z_{2})+D_{2}z_{2}\), where \(D_{1}c_{1}+D_{2}c_{2}=0\) and \(G_{3}(z_{1})\), \(G_{4}(z_{2})\) are finiteorder entire periodic functions in \(z_{1}\), \(z_{2}\) with period \(c_{1}\), \(c_{2}\), respectively.
From the definitions of \(\gamma _{1}(z)\) and \(\gamma _{2}(z)\) we can see that
where \(L(z)=L_{1}(z)+L_{2}(z)\), \(B=B_{1}+B_{2}\).
Subcase 2.2. Suppose that \(Q_{6}(z)e^{\gamma _{2}(z)\gamma _{1}(z+c)}\equiv 1\). Similarly to the argument in Subcase 2.2 in Theorem 2.1, we can get a contradiction.
This completes the proof of Theorem 2.4.
Proof of Theorem 2.3
Similar to the argument in the proof of Theorem 2.1, we can easily prove the statements of Theorem 2.3.
Remarks
In view of the arguments in the proofs of Theorems 2.1 and 2.3, we easily get the following theorems.
Theorem 7.1
Let \(c = (c_{1}, c_{2})\in \mathbb{C}^{2}\) with \(c_{1}\neq 0\), \(c_{2}\neq 0\), and \(c_{1}+c_{2}\neq 0\). If the partial differentialdifference equation
admits a transcendental entire solution \(f(z_{1},z_{2})\) of finite order, then \(g(z)\) must be a polynomial function of the form \(g(z)=L(z)+B\), where \(L(z)\) is a linear function of the form \(L(z)=A_{1}z_{1}+A_{2}z_{2}+B\), \(A_{1}, A_{2}, B\in \mathbb{C}\). Further, \(f(z_{1},z_{2})\) must satisfy one of the following cases:

(i)
$$ f(z_{1},z_{2})=\frac{4(\xi ^{2}+1)}{A_{1} (A_{1}+A_{2})\xi } e^{ \frac{1}{2}g(z_{1},z_{2})} $$
with \(\xi (\neq 0)\), \(A_{1}, A_{2}, B\in \mathbb{C}\) satisfying
$$ \frac{1}{4}\frac{\xi ^{2}1}{(\xi ^{2}+1)i}A_{1} (A_{1}+A_{2})=e^{ \frac{1}{2}(A_{1}c_{1}+A_{2}c_{2})}; $$ 
(ii)
$$ f(z_{1},z_{2})=\frac{e^{L_{1}(z)+B_{1}}}{2A_{11}(A_{11}+A_{12})}+ \frac{e^{L_{2}(z)+B_{2}}}{2A_{21}(A_{21}+A_{12})}, $$
where \(L_{1}(z)=A_{11}z_{1}+A_{12}z_{2}+B_{1}\), \(L_{2}(z)=A_{21}z_{1}+A_{22}z_{2}+B_{2}\), \(A_{j1},A_{j2},B_{j}\in \mathbb{C}\) (\(j=1,2\)) satisfy
$$ L_{1}(z)\neq L_{2}(z),\qquad g(z)=L_{1}(z)+L_{2}(z)+B_{1}+B_{2}, $$and
$$ iA_{11}(A_{11}+A_{12})e^{L_{1}(c)}=iA_{21}(A_{21}+A_{22})e^{L_{2}(c)}=1. $$
We give some examples showing the existence of finiteorder transcendental entire solutions of equation (7.1).
Example 7.1
Let \(A_{1}=1\), \(A_{2}=1\), \(B=0\), and
Then \(\rho (f)=1\), and \(f(z_{1},z_{2})\) is a transcendental entire solution of equation (7.1) with \(g(z)=z_{1}+z_{2}\), \(c_{1}=2\pi i\), and \(c_{2}=2\pi i\).
Example 7.2
Let \(L_{1}(z)=iz_{1}+(1i)z_{2}\), \(L_{2}(z)=z_{1}+(i1)z_{2}\), \(B_{1}=B_{2}=0\), and
Then \(\rho (f)=1\), and \(f(z_{1},z_{2})\) is a transcendental entire solution of equation (7.1) with \(g(z)=(1+i)z_{1}\), \(c_{1}=\frac{3\pi }{2}(1+i)\), and \(c_{2}=\frac{\pi }{2}(1+2i)\).
Theorem 7.2
Let \(c = (c_{1}, c_{2})\in \mathbb{C}^{2}\) with \(c_{1}\neq 0\), \(c_{2}\neq 0\), and \(c_{1}+c_{2}\neq 0\). If the partial differentialdifference equation
admits a transcendental entire solution \(f(z_{1},z_{2})\) of finite order, then \(g(z)\) must be a polynomial function of the form \(g(z)=L(z)+B\), where \(L(z)\) is a linear function of the form \(L(z)=A_{1}z_{1}+A_{2}z_{2}+B\), \(A_{1}, A_{2}, B\in \mathbb{C}\). Further, \(f(z_{1},z_{2})\) must satisfy one of the following cases:

(i)
$$ f(z_{1},z_{2})=\frac{4(\xi ^{2}+1)}{(A_{1}^{2}+A_{2}^{2})\xi } e^{ \frac{1}{2}g(z_{1},z_{2})} $$
with \(\xi (\neq 0)\), \(A_{1}, A_{2}, B\in \mathbb{C}\) satisfying
$$ \frac{1}{4}\frac{\xi ^{2}1}{(\xi ^{2}+1)i} \bigl(A_{1}^{2}+A_{2}^{2} \bigr)=e^{ \frac{1}{2}(A_{1}c_{1}+A_{2}c_{2})}; $$ 
(ii)
$$ f(z_{1},z_{2})=\frac{e^{L_{1}(z)+B_{1}}}{2(A_{11}^{2}+A_{12}^{2})}+ \frac{e^{L_{2}(z)+B_{2}}}{2(A_{21}^{2}+A_{12}^{2})}, $$
where \(L_{1}(z)=A_{11}z_{1}+A_{12}z_{2}+B_{1}\), \(L_{2}(z)=A_{21}z_{1}+A_{22}z_{2}+B_{2}\), \(A_{j1},A_{j2},B_{j}\in \mathbb{C}\) (\(j=1,2\)) satisfy
$$ L_{1}(z)\neq L_{2}(z),\qquad g(z)=L_{1}(z)+L_{2}(z)+B_{1}+B_{2}, $$and
$$ i\bigl(A_{11}^{2}+A_{12}^{2} \bigr)e^{L_{1}(c)}=i\bigl(A_{21}^{2}+A_{22}^{2} \bigr)e^{L_{2}(c)}=1. $$
Some examples explain the existence of finiteorder – entire solutions of equation (7.2).
Example 7.3
Let \(A_{1}=1\), \(A_{2}=1\), \(B=0\), and
Then \(\rho (f)=1\), and \(f(z_{1},z_{2})\) is a transcendental entire solution of equation (7.2) with \(g(z)=z_{1}+z_{2}\), \(c_{1}=\pi i\), and \(c_{2}=\pi i\).
Example 7.4
Let \(L_{1}(z)=iz_{1}+\sqrt{2}z_{2}\), \(L_{2}(z)=\sqrt{2}iz_{1}+z_{2}\), \(B_{1}=B_{2}=0\), and
Then \(\rho (f)=1\) and \(f(z_{1},z_{2})\) is a transcendental entire solution of equation (7.2) with \(g(z)=(\sqrt{2}+1)iz_{1}+(\sqrt{2}+1)z_{2}\), \(c_{1}=\frac{(\sqrt{2}3)\pi }{2}\), and \(c_{2}=\frac{(3\sqrt{2}1)}{2}\pi i\).
In view of Theorems 7.1 and 7.2, we easily get the following.
Corollary 7.1
Let \(c = (c_{1}, c_{2})\in \mathbb{C}^{2}\), \(c_{1}\neq 0\), \(c_{2}\neq 0\), \(\alpha ,\beta ,\gamma \in \mathbb{C}\), and let \(g(z_{1},z_{2})\) be not a linear function of the form \(L(z)=A_{1}z_{1}+A_{2}z_{2}+B\), where \(A_{1},A_{2},B\in \mathbb{C}\). If \(\alpha c_{1}^{2}\beta c_{1}c_{2}+\gamma c_{2}^{2}\neq 0\), then the partial differentialdifference equation
has no finiteorder transcendental entire solution.
Corollary 7.2
The finiteorder transcendental entire solution \(f(z_{1},z_{2})\) of the partial differential equations
must be of the form
where \(L(z)=A_{1}z_{1}+A_{2}z_{2}\), \(A_{1},A_{2},B\in \mathbb{C}\) satisfy \(A_{1}^{2}(A_{1}+A_{2})^{2}=1\) and \((A_{1}^{2}+A_{2}^{2})^{2}=1\).
Corresponding to Theorems 7.1 and 7.2, we can obtain some results on the existence of solutions of the differencetype equations (7.1) and (7.2).
Theorem 7.3
Let \(c = (c_{1}, c_{2})\in \mathbb{C}^{2}\), \(c_{2}\neq 0\), \(c_{1}\neq c_{2}\). If the partial differentialdifference equation
admits a transcendental entire solution of finite order, then \(g(z_{1},z_{2})\) must be a linear function of the form \(g(z_{1},z_{2})=A_{1}z_{1}+A_{2}z_{2}+B\), where \(A_{1}, A_{2}, B\in \mathbb{C}\).
Theorem 7.4
Let \(c = (c_{1}, c_{2})\) be a constant in \(\mathbb{C}^{2}\) such that \(c_{1}\neq \pm ic_{2}\). If the partial differentialdifference equation
admits a transcendental entire solution of finite order, then \(g(z_{1},z_{2})\) must be a linear function of the form \(g(z_{1},z_{2})=A_{1}z_{1}+A_{2}z_{2}+B\), where \(A_{1}, A_{2}, B\in \mathbb{C}\).
Remark 7.1
Although we give the conditions for the existence of finiteorder transcendental entire solutions of equations (7.3) and (7.4) in Theorems 7.3 and 7.4, in view of Theorems 2.2 and 2.4, there naturally arises an open question: How to describe the forms of finiteorder transcendental entire solutions of equations (7.3) and (7.4)?
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References
 1.
Xu, L., Cao, T.B.: Solutions of complex Fermattype partial difference and differentialdifference equations. Mediterr. J. Math. 15, 1–14 (2018)
 2.
Xu, L., Cao, T.B.: Correction to: solutions of complex Fermattype partial difference and differentialdifference equations. Mediterr. J. Math. 17, 1–4 (2020)
 3.
Liu, K.: Meromorphic functions sharing a set with applications to difference equations. J. Math. Anal. Appl. 359, 384–393 (2009)
 4.
Liu, K., Cao, T.B.: Entire solutions of Fermat type difference differential equations. Electron. J. Differ. Equ. 2013, 59 (2013)
 5.
Liu, K., Cao, T.B., Cao, H.Z.: Entire solutions of Fermat type differentialdifference equations. Arch. Math. 99, 147–155 (2012)
 6.
Gross, F.: On the equation \(f^{n}+g^{n} = 1\). Bull. Am. Math. Soc. 72, 86–88 (1966)
 7.
Montel, P.: Leçons sur les Familles Normales de Fonctions Analytiques et Leurs Applications. GauthierVillars, Paris (1927)
 8.
Han, Q., Lü, F.: On the equation \(f^{n}(z)+g^{n}(z) = e^{\alpha z+\beta}\). J. Contemp. Math. Anal. 54, 98–102 (2019)
 9.
Lü, F., Lü, W.R., Li, C.P., Xu, J.F.: Growth and uniqueness related to complex differential and difference equations. Results Math. 74, 30 (2019)
 10.
Liu, K., Song, C.J.: Meromorphic solutions of complex differentialdifference equations. Results Math. 72, 1759–1771 (2017)
 11.
Naftalevich, A.: On a differentialdifference equation. Mich. Math. J. 19, 59–65 (1966)
 12.
Naftalevich, A., Gylys, A.: On meromorphic solutions of a linear differentialdifference equation with constant coefficients. Mich. Math. J. 27, 195–213 (1980)
 13.
Qi, X.G., Liu, Y., Yang, L.Z.: A note on solutions of some differentialdifference equations. J. Contemp. Math. Anal. (Armen. Acad. Sci.) 52, 128–133 (2017)
 14.
Qi, X.G., Yang, L.Z.: Entire solutions of some differentialdifference equations. Bull. Iran. Math. Soc. 52, 1–12 (2020)
 15.
Zhang, J.: On some special difference equations of Malmquist type. Bull. Korean Math. Soc. 55, 51–61 (2018)
 16.
Chiang, Y.M., Feng, S.J.: On the Nevanlinna characteristic of \(f(z+\eta)\) and difference equations in the complex plane. Ramanujan J. 16, 105–129 (2008)
 17.
Halburd, R.G., Korhonen, R.J.: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 314, 477–487 (2006)
 18.
Halburd, R.G., Korhonen, R.J.: Finiteorder meromorphic solutions and the discrete Painlevé equations. Proc. Lond. Math. Soc. 94, 443–474 (2007)
 19.
Halburd, R.G., Korhonen, R.J.: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn., Math. 31, 463–478 (2006)
 20.
Liu, M.L., Gao, L.Y.: Transcendental solutions of systems of complex differentialdifference equations. Sci. Sin., Math. 49, 1–22 (2019) (in Chinese)
 21.
Hu, P.C., Li, B.Q.: On meromorphic solutions of nonlinear partial differential equations of first order. J. Math. Anal. Appl. 377, 881–888 (2011)
 22.
Khavinson, D.: A note on entire solutions of the eiconal equation. Am. Math. Mon. 102, 159–161 (1995)
 23.
Li, B.Q.: On entire solutions of Fermat type partial differential equations. Int. J. Math. 15, 473–485 (2004)
 24.
Lü, F., Li, Z.: Meromorphic solutions of Fermat type partial differential equations. J. Math. Anal. Appl. 478, 864–873 (2019)
 25.
Saleeby, E.G.: Entire and meromorphic solutions of Fermat type partial differential equations. Analysis 19, 369–376 (1999)
 26.
Courant, R., Hilbert, D.: Methods of Mathematical Physics, Vol. II, Partial Differential Equations. Interscience, New York (1962)
 27.
Garabedian, P.R.: Partial Differential Equations. Wiley, New York (1964)
 28.
Li, B.Q.: Entire solutions of \((u_{z_{1}})^{m}+(u_{z_{2}})^{n}=e^{g}\). Nagoya Math. J. 178, 151–162 (2005)
 29.
Cao, T.B., Xu, L.: Logarithmic difference lemma in several complex variables and partial difference equations. Ann. Mat. 199, 767–794 (2020)
 30.
Cao, T.B., Korhonen, R.J.: A new version of the second main theorem for meromorphic mappings intersecting hyperplanes in several complex variables. J. Math. Anal. Appl. 444, 1114–1132 (2016)
 31.
Korhonen, R.J.: A difference Picard theorem for meromorphic functions of several variables. Comput. Methods Funct. Theory 12, 343–361 (2012)
 32.
Ronkin, L.I.: Introduction to the Theory of Entire Functions of Several Variables. Nauka, Moscow (1971) (Russian)
 33.
Stoll, W.: Holomorphic Functions of Finite Order in Several Complex Variables. Am. Math. Soc., Providence (1974)
 34.
Pólya, G.: On an integral function of an integral function. J. Lond. Math. Soc. 1, 12–15 (1926)
 35.
Hu, P.C., Li, P., Yang, C.C.: Unicity of Meromorphic Mappings, Advances in Complex Analysis and Its Applications, vol. 1. Kluwer Academic, Dordrecht (2003)
Acknowledgements
We thank the referee(s) for reading the manuscript very carefully and making a number of valuable and kind comments, which improved the presentation.
Funding
This work was supported by the National Natural Science Foundation of China (11561033, 61877046), the Natural Science Foundation of Jiangxi Province in China (20181BAB201001), Shangrao Science and Technology Talent Plan (2020K006) and the Foundation of Education Department of Jiangxi (GJJ190876, GJJ202303, GJJ191042, GJJ190895) of China.
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Xu, H.Y., Meng, D.W., Liu, S. et al. Entire solutions for several secondorder partial differentialdifference equations of Fermat type with two complex variables. Adv Differ Equ 2021, 52 (2021). https://doi.org/10.1186/s1366202003201y
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MSC
 30D35
 35M30
 32W50
 39A45
Keywords
 Nevanlinna theory
 Existence
 Entire solution
 Partial differentialdifference equation