Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 18, pp. 1–11. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXACT FORMS OF ENTIRE SOLUTIONS FOR FERMAT TYPE PARTIAL DIFFERENTIAL EQUATIONS IN C2 YU XIAN CHEN, HONG YAN XU Abstract. This article studies the existence and the exact form of entire so- lutions of several Fermat type partial differential equations in C2, by utilizing the Nevanlinna theory of meromorphic functions in several complex variables. We obtain results about the existence and form of transcendental entire solu- tions with finite order for some variations of Fermat type functional equations. Our results are extensions and generalizations of the previous theorems by Xu and Cao [29, 30], Liu and Dong [19]. 1. Introduction and statement of main results In 1939, Iyer [10] studied solutions of the Fermat type functional equation f2(z) + g2(z) = 1, (1.1) and proved the classical result that the entire solutions of equation (1.1) are f = cos a(z), g = sin a(z), where a(z) is an entire function, no other solutions exist. After his work, many scholars had paid considerable attention to the existence and the form of entire and meromorphic solutions of some variations of (1.1); for details, we refer readers to [8, 25, 31, 32]. In 2004, Yang and Li [31] discussed the form of solutions of the equations, where g(z) is replaced by f ′(z) in (1.1), that is, f2(z) + (f ′(z))2 = 1, (1.2) they proved that (1.2) has only transcendental entire solutions of the form f(z) = 1 2 ( Peαz + 1 P e−αz ) , where P, α are nonzero constants. They also studied the existence of solutions of the equation when f ′(z) is replaced by a differential polynomial in f and obtained the following theorem. Theorem 1.1 ([31, Theorem 2]). Let bn and bn+1 be nonzero constants. Then f2(z) + [bnf (n)(z) + bn+1f (n+1)(z)]2 = 1 (1.3) has no transcendental meromorphic solutions. 2010 Mathematics Subject Classification. 30D35, 35M30, 32W50, 39A45. Key words and phrases. Fermat type; Nevanlinna theory; existence; entire solution; complex partial differential equation. c©2021 Texas State University. Submitted January 7, 2021. Published March 24, 2021. 1 2 Y. X. CHEN, H. Y. XU EJDE-2021/18 In 2015, Liu and Dong [19] further investigated the existence of solutions of the Fermat type equation (1.1), when both f(z) and g(z) are replaced by differential polynomials in f(z). they proved the following result. Theorem 1.2 ([19, Theorem 1.7]). The equation [f(z) + f ′(z)]2 + [f(z) + f ′′(z)]2 = 1 (1.4) has no transcendental meromorphic solutions. It is always an interesting and quite difficult problem to prove the existence and the form of the entire or meromorphic solution of differential equation in the com- plex plane C. In the past five or more decades, Nevanlinna theory of meromorphic functions has been used widely to deal with these problems and derive many inter- esting results of meromorphic solutions of differential equations in complex plane (see, e.g.,[1, 13, 17]). Especially, Yang [33], Yi and Yang [34], and Li and Yang [16] studied the existence and the form of the entire and meromorphic solutions of complex Fermat type differential equations in C, by employing the Nevanlinna theory. Very recently, with the development of the Nevanlinna theory with several com- plex variables (see [2, 3, 12]), Xu and Cao [29, 30], Xu and coauthors [27, 28] investigated the existence of solutions for some Fermat type partial differential equations with two complex variables by using the difference logarithmic derivative lemma of several complex variables, and extended the results of Yang and Li [31] from one complex variable to several complex variables. Theorem 1.3 ([30, Corollary 1.4]). Any transcendental entire solution with finite order of the partial differential equation of the Fermat type f2(z1, z2) + (∂f(z1, z2) ∂z1 )2 = 1 (1.5) has the form of f(z1, z2) = sin(z1 + g(z2)), where g(z2) is a polynomial in one variable z2. The study of complex partial differential equations has a long history, see for example [5, 7, 22], and for equations with several complex variables see [9, 11, 15, 20, 22]. Khavinson [11] pointed out that any entire solution of the partial differential equation ( ∂f ∂z1 )2 + ( ∂f ∂z2 )2 = 1 (1.6) in C2 is necessarily linear. This partial differential equations in the real variable case occur in the study of characteristic surfaces and in wave propagation theory, and it is the two dimensional eiconal equation, one of the main equations of geometric optics (see [6, 7]). In 1999, Saleeby [22] studied the entire solution of Fermat type partial differential equation (1.6) and obtain the following result. Theorem 1.4 ([22, Theorem 1]). If f is an entire solution of (1.6) in C2, then f = c1z1 + c2z2 + c, where c1, c2, c ∈ C and c21 + c22 = 1. Later, Li and his coauthors [4, 14, 15] discussed some variations of the partial differential equation (1.6), and obtained interesting and important results, of which we mention the following. EJDE-2021/18 FERMAT TYPE PARTIAL DIFFERENTIAL EQUATIONS 3 Theorem 1.5 ([4, Corollary 2.3]). Let P (z1, z2) and Q(z1, z2) be arbitrary polyno- mials in C2. Then f is an entire solution of the equation( P ∂f ∂z1 )2 + ( Q ∂f ∂z2 )2 = 1 if and only if f = c1z1 + c2z2 + c3 is a linear function, where cj’s are constants, and exactly one of the following holds: (i) c1 = 0 and Q is a constant satisfying that (c2Q)2 = 1; (ii) c2 = 0 and P is a constant satisfying that (c1P )2 = 1; (iii) c1c2 6= 0 and P,Q are both constants satisfying that (c1P )2 + (c2Q)2 = 1. From Theorems 1.1–1.3, a question can be naturally raised: What will happen to the existence and the form of the solutions when equations (1.3) and (1.4) are turned from one complex vari- able to several complex variables? Motivated by this question, this article considers the description of entire solutions for some variations of the partial differential equation (1.5) in more general form. The main tool in this paper is the Nevanlinna theory with several complex variables. Our main results generalize the previous theorems given by Xu and Cao, Liu and Dong [19, 30]. Throughout this article, for convenience, we assume that z + w = (z1 + w1, z2 + w2) for any z = (z1, z2), w = (w1, w2). Firstly, we consider the transcendental entire solution with finite order of the first order partial differential equation of Fermat type,[ a1f(z) + a2 ∂f ∂z1 ]2 + [ a3f(z) + a4 ∂f ∂z2 ]2 = 1, (1.7) where a1, a2, a3, a4 ∈ C. Theorem 1.6. Let a1, a2, a3, a4 ∈ C be four nonzero constants. Then the transcen- dental entire solution f(z1, z2) with finite order of the partial differential equation (1.7) must be of the form f(z1, z2) = ± 1√ a21 + a23 + ηe−( a1 a2 z1+ a3 a4 z2), or f(z1, z2) = a3 + ia1 2(α1a2a3 − α2a1a4) eL(z)+B − a3 − ia1 2(α1a2a3 − α2a1a4) e−L(z)−B + ηe−( a1 a2 z1+ a3 a4 z2), where L(z) = α1z2 + α2z2, α1 = a3 a2 i, α2 = −a1a4 i, and η,B ∈ C. The following example shows that the forms of the solutions in Theorem 1.6 are precise. Example 1.7. Let η ∈ C and η 6= 0, and f(z1, z2) = ± 1√ 5 + ηe−(2z1+z2), g(z1, z2) = 1 + 2i 10i ei(z1−2z2) − 1− 2i 10i e−i(z1−2z2) + ηe−(2z1+z2). Then ρ(f) = ρ(g) = 1 and f(z1, z2), g(z1, z2) are the finite order transcendental entire solutions for (1.7) with a1 = 2, a2 = a3 = a4 = 1. 4 Y. X. CHEN, H. Y. XU EJDE-2021/18 From Theorem 1.6, one can easily obtain the following corollary. Corollary 1.8. Let f(z1, z2) be a transcendental entire solution with finite order of the partial differential equation[ f(z) + ∂f ∂z1 ]2 + [ f(z) + ∂f ∂z2 ]2 = 1. (1.8) Then f(z1, z2) is of the form f(z1, z2) = ± √ 2 2 + ηe−(z1+z2), or f(z1, z2) = sin(z2 − z1 + η1)− cos(z2 − z1 + η1) + η2e −(z1+z2), where η, η1, η2 ∈ C. Secondly, we study the existence and the form of transcendental entire solutions of several second order partial differential equations of Fermat type,[ a1f(z) + a2 ∂f ∂z1 ]2 + [ a3f(z) + a4 ∂2f ∂z21 ]2 = 1, (1.9) and ( a2 ∂f ∂z1 )2 + [ a3f(z) + a4 ∂2f ∂z21 ]2 = 1, (1.10) where a1, a2, a3, a4 ∈ C. Theorem 1.9. Let a1, a2, a3, a4 ∈ C be four nonzero constants such that D := −(a21a4 +a22a3) 6= 0. Then the partial differential equation (1.9) does not admit any transcendental entire solution with finite order. From Theorem 1.9, we have the following corollary. Corollary 1.10. The partial differential equation[ f(z) + ∂f ∂z1 ]2 + [ f(z) + ∂2f ∂z21 ]2 = 1 does not admit any transcendental entire solution with finite order. For a1 = 0 in (1.9), we have the following result. Theorem 1.11. Let a2, a3, a4 ∈ C be three nonzero constants. Then (1.10) admits any transcendental entire solution f(z1, z2) with finite order, and f(z1, z2) must be of the form f(z1, z2) = −α1a4 + ia2 a2a3 sh(α1z1 + ϕ(z2)), where ϕ(z2) is a polynomial in z2, and α1 = (−a2 ± √ a22 + 4a3a4)i 2a4 . Similar to the above argument, we discuss the transcendental entire solutions of some second mix partial differential equations. We obtain the following theorem. Theorem 1.12. Let a2, a3, a4 be three nonzero constants and a1 ∈ C, and[ a1f(z) + a2 ∂f ∂z1 ]2 + [ a3f(z) + a4 ∂2f ∂z1∂z2 ]2 = 1. (1.11) Then EJDE-2021/18 FERMAT TYPE PARTIAL DIFFERENTIAL EQUATIONS 5 (i) if a1 6= 0, then equation (1.11) has no any transcendental entire solution with finite order; (ii) if a1 = 0, then the finite order transcendental entire solution f(z1, z2) of equation (1.11) must be of the form f(z1, z2) = −α1a4 + ia2 a2a3 sh(α1z1 + α2z2 +B), where α1, α2, B are constants and satisfy α1 = − a3 α2a4+a2i . The following example shows the existence of a transcendental entire solution of equation (1.11). Example 1.13. Let f(z1, z2) = − (1 + √ 2)i 4 ( e( √ 2−1)iz1+zn2 − e−[( √ 2−1)iz1+zn2 ] ) , n ∈ N+. Then ρ(f) = n and f(z1, z2) is a finite order transcendental entire solution for (1.11) with a2 = 2, a3 = a4 = 1 and α1 = ( √ 2− 1)i. From Theorem 1.9, we can easily obtain the following corollary. Corollary 1.14. The partial differential equation[ f(z) + ∂f ∂z1 ]2 + [ f(z) + ∂2f ∂z1∂z2 ]2 = 1 does not admit any transcendental entire solution with finite order. Finally, we can obtain the following results by using the same arguments as in Theorem 1.8. Theorem 1.15. Let b1 and b2 be two nonzero constants in C. Then f2(z) + [ b1 ∂f ∂z1 + b2 ∂2f ∂z21 ]2 = 1 (1.12) has no finite order transcendental entire solutions. Theorem 1.16. Let b1 and b2 be two nonzero constants in C. Then the finite order transcendental entire solution f(z1, z2) of equation f2(z) + [ b1 ∂f ∂z1 + b2 ∂2f ∂z1∂z2 ]2 = 1 (1.13) must be of the form f = sin ( 1 b1 z1 + η ) , where η ∈ C. Theorems 1.15 and 1.16 are extensions of Theorem 1.1 from one complex variable to two complex variables. 2. Proof of Theorem 1.6 The following lemmas play the key roles in proving our results. Lemma 2.1 ([23, 24]). For an entire function F on Cn, with F (0) 6= 0 and ρ(nF ) = ρ < ∞. Then there exists a canonical function fF and a function gF ∈ Cn such that F (z) = fF (z)egF (z). For the special case n = 1, fF is the canonical product of Weierstrass. Here, ρ(nF ) denotes the order of the counting function of zeros of F . 6 Y. X. CHEN, H. Y. XU EJDE-2021/18 Lemma 2.2 ([21]). . If g and h are entire functions on the complex plane C and g(h) is an entire function of finite order, then there are only two possible cases: either (a) the internal function h is a polynomial and the external function g is of finite order; or (b) the internal function h is not a polynomial but a function of finite order, and the external function g is of zero order. Prof of Theorem 1.6. Suppose that f(z) is a transcendental entire solution with finite order of (1.7). Two cases will be discussed below. Case 1: f(z) + ∂f ∂z1 is a constant. We set a1f(z) + a2 ∂f ∂z1 = K1, K1 ∈ C. (2.1) In view of (1.7), it follows that a3f(z) + a4 ∂f ∂z2 is a constant, let a3f(z) + a4 ∂f ∂z2 = K2, K2 ∈ C. (2.2) This leads to K2 1 +K2 2 = 1. In view of (2.1) and (2.2), it follows that a2a3 ∂f ∂z1 − a1a4 ∂f ∂z2 = a3K1 − a1K2. (2.3) The characteristic equations of (2.3) are dz1 dt = a2a3, dz2 dt = −a1a4, df dt = a3K1 − a1K2. Using the initial conditions: z1 = 0, z2 = s, and f = f(0, s) := φ(s) with a parame- ter s. Thus, we obtain the following parametric representation for the solutions of the characteristic equations: z1 = a2a3t, z2 = −a1a4t+ s, f(t, s) = ∫ t 0 a3K1 − a1K2dt+ φ(s) = (a3K1 − a1K2)t+ φ(s), where φ(s) is a transcendental entire function with finite order in s. Noting that t = z1 a2a3 and s = z2 + a1a4 a2a3 z1, then the solution of (2.3) is of the form f(z1, z2) = (a3K1 − a1K2) z1 a2a3 + φ(z2 + a1a4 a2a3 z1). (2.4) On the other hand, differentiating both two sides of the equations (2.1), (2.2) for the variables z2, z1, respectively, and noting the fact that ∂2f ∂z1∂z2 = ∂2f ∂z2∂z1 , it follows that a2a3 ∂f ∂z1 = a1a4 ∂f ∂z2 , which implies that a3K1 = a1K2. Thus, it follows that K1 = ± a1√ a21 + a23 , K2 = ± a3√ a21 + a23 , f(z1, z2) = φ(z2 + a1a4 a2a3 z1). Substituting these into (2.2) and (2.3), we obtain φ(z2 + a1a4 a2a3 z1) + a4 a3 φ′(z2 + a1a4 a2a3 z1) = ± 1√ a21 + a23 . EJDE-2021/18 FERMAT TYPE PARTIAL DIFFERENTIAL EQUATIONS 7 This means that f(z1, z2) = φ(z2 + a1a4 a2a3 z1) = ± 1√ a21 + a23 + ηe−( a1 a2 z1+ a3 a4 z2). Case 2: a1f(z) + a2 ∂f ∂z1 is not a constant. From the fact that the entire solutions of equation f2 + g2 = 1 are f = cos a(z), g = sin a(z), we can deduce that a1f(z) + a2 ∂f ∂z1 is transcendental, where a(z) is an entire function. Thus, we rewrite (1.7) in the form [ a1f(z) + a2 ∂f ∂z1 + i ( a3f(z) + a4 ∂f ∂z2 )] × [ a1f(z) + a2 ∂f ∂z1 − i ( a3f(z) + a4 ∂f ∂z2 )] = 1, (2.5) which implies that both a1f + a2 ∂f ∂z1 + i ( a3f + a4 ∂f ∂z2 ) and a1f + a2 ∂f ∂z1 − i ( a3f + a4 ∂f ∂z2 ) have no poles and zeros. Thus, by Lemmas 2.1 and 2.2, there thus exists a polynomial p(z) such that a1f(z) + a2 ∂f ∂z1 + i ( a3f(z) + a4 ∂f ∂z2 ) = ep(z), a1f(z) + a2 ∂f ∂z1 − i ( a3f(z) + a4 ∂f ∂z2 ) = e−p(z), which leads to a1f(z) + a2 ∂f(z) ∂z1 = ep(z) + e−p(z) 2 , (2.6) a3f(z) + a4 ∂f(z) ∂z2 = ep(z) − e−p(z) 2i . (2.7) This means that a2a3 ∂f(z) ∂z1 − a1a4 ∂f(z) ∂z2 = a3 + ia1 2 ep(z) + a3 − ia1 2 e−p(z). (2.8) Differentiating on z2, z1 for both two sides of equations (2.6), (2.7), respectively, and noting the fact that ∂2f ∂z1∂z2 = ∂2f ∂z2∂z1 , we can conclude that a2a3 ∂f ∂z1 −a1a4 ∂f ∂z2 = −1 2 (ia2 ∂p ∂z1 +a4 ∂p ∂z2 )ep(z)+ 1 2 (−ia2 ∂p ∂z1 +a4 ∂p ∂z2 )e−p(z). (2.9) Thus, it follows from (2.8) and (2.9) that e2p ( a2i ∂p ∂z1 + a4 ∂p ∂z2 + a1i+ a3 ) = −a2i ∂p ∂z1 + a4 ∂p ∂z2 + a1i− a3. (2.10) Suppose that a2i ∂p ∂z1 + a4 ∂p ∂z2 + a1i+ a3 6= 0 and −a2i ∂p∂z1 + a4 ∂p ∂z2 + a1i− a3 6= 0. Since f(z) is a finite order transcendental entire solution of equation (1.7), by Lemma 2.1, 2.2 and (2.10), we conclude that p(z) is a nonconstant polynomial in C2. Thus, a contradiction can be obtained from (2.10) using Nevanlinna theory. In fact, if T (r, F ) denotes the Nevanlinna characteristic function of a meromorphic function F in C2, then by (2.10) we deduce that T (r, e2p) = O{T (r, p) + log r}, outside possibly a set of finite Lebesgue measure, using the results (see e.g. [26, p.99], [24]) that T (r, Fzj ) = O{T (r, F )} for any meromorphic function F outside a set of finite Lebesgue measure and that T (r, P ) = O{log r} for any polynomial P . 8 Y. X. CHEN, H. Y. XU EJDE-2021/18 But, limr→∞ T (r,e2p) T (r,p)+log r = +∞ when p is a nonconstant polynomial. Therefore, p must be constant, a contradiction. Thus, equation (2.10) implies that a2i ∂p ∂z1 + a4 ∂p ∂z2 + a1i+ a3 = 0, −a2i ∂p ∂z1 + a4 ∂p ∂z2 + a1i− a3 = 0. Hence, it follows that α1 := ∂p ∂z1 = a3 a2 i and α2 := ∂p ∂z2 = −a1a4 i, which means that p(z1, z2) = α1z1 + α2z2 + η1 = a3 a2 iz1 − a1 a4 iz2 +B where B ∈ C. On the other hand, it follows from (2.8) that a2a3 ∂f(z) ∂z1 − a1a4 ∂f(z) ∂z2 = a3 + ia1 2 eα1z1+α2z2+B + a3 − ia1 2 e−(α1z1+α2z2+B). Then the characteristic equations for this differential equation are dz1 dt = a2a3, dz2 dt = −a1a4, df dt = a3 + ia1 2 eα1z1+α2z2+B + a3 − ia1 2 e−(α1z1+α2z2+B). Using the initial conditions: z1 = 0, z2 = s, and f = f(0, s) := ϕ0(s) with a param- eter s. Thus, we obtain the following parametric representation for the solutions of the characteristic equations: z1 = a2a3t, z2 = −a1a4t+ s, f(t, s) = ∫ t 0 (1 + i 2 e(α1a2a3−α2a1a4)t+α2s+B + 1− i 2 e−[(α1a2a3−α2a1a4)t+α2s+B] ) dt+ ϕ0(s) = a3 + ia1 2(α1a2a3 − α2a1a4) e(α1a2a3−α2a1a4)t+α2s+B − a3 − ia1 2(α1a2a3 − α2a1a4) e−[(α1a2a3−α2a1a4)t+α2s+B] + ϕ(s), where ϕ(s) is an entire function with finite order in s such that ϕ(s) = ϕ0(s)− a3 + ia1 2(α1a2a3 − α2a1a4) eα2s+B + a3 − ia1 2(α1a2a3 − α2a1a4) e−(α2s+B). Thus, it follows that f(z1, z2) = a3 + ia1 2(α1a2a3 − α2a1a4) eL(z)+B − a3 − ia1 2(α1a2a3 − α2a1a4) e−L(z)−B + ϕ(s). Substituting this expression into (2.6), we can deduce that ϕ(s) satisfies a4 a3 ϕ′(s) + ϕ(s) = 0, (2.11) which implies that φ(s) = ηe−( a1 a2 z1+ a3 a4 z2). Therefore, from Case 1 and Case 2, the proof of Theorem 1.6 is complete. � 3. Proofs of Theorems 1.9–1.12 Proof of Theorem 1.9. Suppose that f(z) is a transcendental entire solution with finite order of (1.9). By using the same argument as in Case 2 of Theorem 1.8, we can easily get that there exists a polynomial p(z) in C2 such that a1f(z) + a2 ∂f ∂z1 = ep(z) + e−p(z) 2 , (3.1) EJDE-2021/18 FERMAT TYPE PARTIAL DIFFERENTIAL EQUATIONS 9 a3f(z) + a4 ∂2f ∂z21 = ep(z) − e−p(z) 2i . (3.2) Thus, the partial derivative of (3.1) for z1 is a1 ∂f ∂z1 + a2 ∂2f ∂z21 = ∂p ∂z1 ep(z) − e−p(z) 2 . (3.3) By combining (3.2) with (3.3), it follows that a2a3f(z)− a1a4 ∂f ∂z1 = ep(z) − e−p(z) 2 (−a22i− ∂p ∂z1 a2a4). (3.4) In view of D := −(a21a4 + a22a3) 6= 0, and by combining with (3.1) and (3.4), we have f(z) = a22i+ a2a4 ∂p ∂z1 − a1a4 2D ep(z) − a22i+ a2a4 ∂p ∂z1 + a1a4 2D e−p(z), (3.5) ∂f ∂z1 = − a1a2i+ a1a4 ∂p ∂z1 + a2a3 2D ep(z) + a1a2i+ a1a4 ∂p ∂z1 − a2a3 2D e−p(z). (3.6) Obviously, p(z) is a nonconstant polynomial. Otherwise, f(z) is a constant, this is a contradiction with the assumption. And in view of (3.5) and (3.6), it follows that (β + γ)e2p(z) = β − γ, (3.7) where β = a1a2i+ a2a4 ∂2p ∂z21 , γ = a2a4 ( ∂p ∂z1 )2 + a22i ∂p ∂z1 + a2a3. Similar to the argument as in the proof of Theorem 1.6, it follows that β + γ = 0 and β − γ = 0, which implies that β = 0 and γ = 0. In view of γ = 0 and a2 6= 0, it follows that a4 ( ∂p ∂z1 )2 + a2i ∂p ∂z1 + a3 = 0, which leads to ∂2p ∂z21 = 0 or ∂p ∂z1 = −2ia4a2 . Combining this with β = 0, we have a1 = 0, this is a contradiction with a1 6= 0. This completes the proof. � Proof of Theorem 1.11. Suppose that f(z) is a transcendental entire solution with finite order of (1.10). By using the same argument as in the proof of Theorem 1.9, we can easily obtain that there exists a nonconstant polynomial p(z) in C2 such that f(z) = − a2i+ a4 ∂p ∂z1 2a2a3 (ep(z) − e−p(z)), ∂f ∂z1 = ep(z) + e−p(z) 2a2 , (3.8) and[ a4 ∂2p ∂z21 +a4 ( ∂p ∂z1 )2 +a2i ∂p ∂z1 +a3 ] e2p(z) = a4 ∂2p ∂z21 −a4 ( ∂p ∂z1 )2 −a2i ∂p ∂z1 −a3. (3.9) Thus, it follows that a4 ∂2p ∂z21 +a4 ( ∂p ∂z1 )2 +a2i ∂p ∂z1 +a3 = 0, a4 ∂2p ∂z21 −a4 ( ∂p ∂z1 )2 −a2i ∂p ∂z1 −a3 = 0. (3.10) Hence, it means that a4 ∂2p ∂z21 = 0 and a4 ( ∂p ∂z1 )2 + a2i ∂p ∂z1 + a3 = 0. Since a2, a3, a4 are nonzero constants, it follows that ∂2p ∂z21 = 0 and ∂p ∂z1 is a constant and a root of the equation a4ω 2 + a2iω+ a3 = 0. Set α1 = ∂p ∂z1 , then α1 = (−a2± √ a22+4a3a4)i 2a4 and 10 Y. X. CHEN, H. Y. XU EJDE-2021/18 p(z) = α1z1 + ϕ(z2), where ϕ(z2) is a polynomial in z2. Substituting these into (3.8), we have f(z1, z2) = −α1a4 + ia2 a2a3 sh(α1z1 + ϕ(z2)). This completes the proof. � The proof of Theorem 1.12 follows the same argument as that of Theorems 1.9 and 1.11; we omit it. 3.1. Acknowledgements. This work was supported by the National Natural Sci- ence Foundation of China (11561033), the Natural Science Foundation of Jiangxi Province in China (20181BAB201001), the Foundation of Education Department of Jiangxi (GJJ202303, GJJ191042, GJJ190876, GJJ190895, GJJ201813) of China, and the Shangrao Science and Technology Talent Plan (2020K006). References [1] T. B. Cao; The growth, oscillation and fixed points of solutions of complex linear differential equations in the unit disc, J. Math. Anal. Appl., 352(2) (2009), 739–748. [2] T. B. Cao, R. J. Korhonen; A new version of the second main theorem for meromorphic mappings intersecting hyperplanes in several complex variables, J. Math. Anal. Appl., 444(2) (2016), 1114–1132. [3] T. B. Cao, L. Xu; Logarithmic difference lemma in several complex variables and partial dif- ference equations, Annali di Matematica (2019). https://doi.org/10.1007/s10231-019-00899- w. [4] D. C. Chang, B. Q. Li; Description of entire solutions of Eiconal type equations, Canad. Math. Bull., 55 (2012), 249–259. [5] S. C. Chen, M. C. Shaw; Partial Differential Equations in Several Complex Domain, Amer- ican Mathematical Society, 2001. [6] R. Courant, D. Hilbert; Methods of Mathematical Physics, Vol II, partial differential equa- tions, Interscience, New York, 1962. [7] P. R. Garabedian; Partial Differential Equations, Wiley, New York, 1964. [8] F. Gross; On the equation fn + gn = 1, Bull. Am. Math. Soc., 72 (1966), 86–88. [9] P. C. Hu, B. Q. Li; On meromorphic solutions of nonlinear partial differential equations of first order, J. Math. Anal. Appl., 377 (2011) 881–888. [10] G. Iyer; On certain functional equations, J. Indian. Math. Soc., 3, (1939), 312–315. [11] D. Khavinson; A note on entire solutions of the eiconal equation, Am. Math. Mon., 102 (1995), 159–161. [12] R. J. Korhonen; A difference Picard theorem for meromorphic functions of several variables, Comput. Methods Funct. Theory, 12(1) (2012), 343–361. [13] I. Laine; Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin/New York, 1993. [14] B. Q. Li; Entire solutions of (uz1 )m + (uz2 )n = eg , Nagoya Math. J., Vol. 178 (2005), 151–162. [15] B. Q. Li; On entire solutions of Fermat type partial differential equations, Int. J. Math., 15 (2004), 473–485. [16] P. Li, C. C. Yang; On the nonexistence of entire solutions of certain type of nonlinear differential equations, J. Math. Anal. Appl., 320 (2006), 827–835. [17] L. W. Liao, C. C. Yang, J. J. Zhang; On meromorphic solutions of certain type of non-linear differential equations, Ann. Acad. Sci. Fenn. Math., 38 (2013), 581–593. [18] K. Liu, T. B. Cao, H. Z. Cao; Entire solutions of Fermat type differential-difference equations, Arch. Math., 99 (2012), 147–155. [19] K. Liu, X. J. Dong; Fermat type differential and difference equations, Electronic Journal of Differential Equations, 2015 (2015), No. 159, 1–10. [20] F. Lü, Z. Li; Meromorphic solutions of Fermat type partial differential equations, J. Math. Anal. Appl., 478 (2) (2019), 864–873. EJDE-2021/18 FERMAT TYPE PARTIAL DIFFERENTIAL EQUATIONS 11 [21] G. Pólya; On an integral function of an integral function, J. Lond. Math. Soc., 1 (1926), 12–15. [22] E. G. Saleeby; Entire and meromorphic solutions of Fermat type partial differential equations, Analysis, 19 (1999), 69-376. [23] L. I. Ronkin; Introduction to the Theory of Entire Functions of Several Variables, Moscow: Nauka 1971 (Russian). American Mathematical Society, Providence, 1974. [24] W. Stoll; Holomorphic Functions of Finite Order in Several Complex Variables, American Mathematical Society, Providence, 1974. [25] J. F. Tang, L. W. Liao; The transcendental meromorphic solutions of a certain type of nonlinear differential equations, J. Math. Anal. Appl., 334 (2007), 517–527. [26] A. Vitter; The lemma of the logarithmic derivative in several complex variables, Duke Math. J., 44 (1977), 89–104. [27] H. Y. Xu, S. Y. Liu, Q.P. Li; Entire solutions for several systems of nonlinear difference and partial differentialdifference equations of fermat-type, Journal of Mathematical Analysis and Applications, 483 (2020), no. 123641, pp. 1–22. [28] H. Y. Xu, H. Wang; Notes on the existence of entire solutions for several partial differential-difference equations, Bulletin of the Iranian Mathematical Society, 46, 2020, http://dx.doi.org/ 10.1007/s41980-020-00453-y. [29] L. Xu, T. B. Cao; Solutions of complex Fermat-type partial difference and differential- difference equations, Mediterr. J. Math., 15 (2018), pages, 1–14. [30] L. Xu, T. B. Cao; Correction to: Solutions of complex Fermat-type partial difference and differential-difference equations, Mediterr. J. Math., 17 (2020), pages, 1–4. [31] C. C. Yang, P. Li; On the transcendental solutions of a certain type of nonlinear differential equations, Arch. Math., 82 (2004), 442–448. [32] L. Z. Yang; Growth of linear differential equations and their applications, Israel J. Math., 147 (2005), 359–372. [33] L. Z. Yang, J. L. Zhang; Non-existence of meromorphic solutions of Fermat type functional equations, Aequationes Mathematices, 76 (2008), 140–150. [34] H. X. Yi, L. Z. Yang; On meromorphic solutions of Fermat type functional equations, Science in China, Ser. A, 41(2011), 907–932. Yu Xian Chen School of Mathematics and computer science, Xinyu University, Xinyu, Jiangxi 338004, China Email address: xygzcyx@126.com Hong Yan Xu (corresponding author) School of Mathematics and computer science, Xinyu University, Xinyu, Jiangxi 338004, China. School of Mathematics and Computer Science, Shangrao Normal University, Shangrao Jiangxi 334001, China Email address: xhyhhh@126.com 1. Introduction and statement of main results 2. Proof of Theorem ?? 3. Proofs of Theorems ??–?? 3.1. Acknowledgements References