Nagoya Mathematical Journal

Entire solutions of $(u_{z_{1}})^{m}+(u_{z_{2}})^{n}=e^{g}$

Bao Qin Li

Full-text: Open access

Abstract

The paper is concerned with description of entire solutions of the partial differential equations $u_{z_{1}}^{m}+u_{z_{2}}^{n}=e^{g}$, where $m \geq 2$, $n \geq 2$ are integers and $g$ is a polynomial or an entire function in ${\bf C}^{2}$. Descriptions are given and complemented by various examples.

Article information

Source
Nagoya Math. J., Volume 178 (2005), 151-162.

Dates
First available in Project Euclid: 16 August 2005

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1124217075

Mathematical Reviews number (MathSciNet)
MR2145319

Zentralblatt MATH identifier
1086.35021

Subjects
Primary: 35F20: Nonlinear first-order equations 32A15: Entire functions 32A22: Nevanlinna theory (local); growth estimates; other inequalities {For geometric theory, see 32H25, 32H30}

Citation

Li, Bao Qin. Entire solutions of $(u_{z_{1}})^{m}+(u_{z_{2}})^{n}=e^{g}$. Nagoya Math. J. 178 (2005), 151--162. https://projecteuclid.org/euclid.nmj/1124217075


Export citation

References

  • C. A. Berenstein and R. Gay, Complex Variables, Springer-Verlag, New York (1991).
  • H. Cartan, Sur les zeros des combinaisons linéaires de $p$ functions holomorphes données , Mathematica (Cluf), 7 (1933), 5–31.
  • R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II, partial differential equations, New York, Interscience (1962).
  • D. C. Chang, B. Q. Li and C. C. Yang, On composition of meromorphic functions in several complex variables , Forum Math., 7 (1995), 77–94.
  • P. R. Garabedian, Partial Differential Equations, New York, Wiley (1964).
  • V. Ganapathy Iyer, On certain functional equations , J. Indian Math. Soc., 3 (1939), 312–315.
  • F. John, Partial Differential Equations, Springer-Verlag, New York (1982).
  • D. Khavinson, A note on entire solutions of the eiconal equation , Amer. Math. Mon., 102 (1995), 159–161.
  • S. Krantz, Function Theory of Several Complex Variables, John Wiley & Sons, New York (1982).
  • J. Markushevich, Entire Functions, Amer. Elsevier Pub. Co., New York (1966).
  • P. Montel, Leçons sur les families normales de functions analytique et leurs applications, Gauthier-Villars, Paris (1927).
  • E. G. Saleeby, Entire and meromorphic solutions of Fermat type partial differential equations , Analysis, 19 (1999), 369–376.
  • B. V. Shabat, Introduction to Complex Analysis, part II, Functions of several variables, Translation Mathematical Monographs, Vol. 110, American Mathematical Society, Providence, RI (1992).
  • W. Stoll, Introduction to the Value Distribution Theory of Meromorphic Functions, Springer-Verlag, New York (1982).
  • A. Vitter, The lemma of the logarithmic derivative in several complex variables , Duke Math. J., 44 (1977), 89–104.