Some results on meromorphic solutions of certain nonlinear differential equations
Bull. Korean Math. Soc.
Published online November 26, 2019
Nan Li and Lianzhong Yang
Qilu Normal University, Shandong University
Abstract : In this paper, we investigate the transcendental meromorphic solutions for the nonlinear differential equations $f^{n}f^{(k)}+Q_{d_{*}}(z,f)=R(z)e^{\alpha(z)}$ and $f^{n}f^{(k)}+Q_{d}(z,f)=p_{1}(z)e^{\alpha_{1}(z)}+p_{2}(z)e^{\alpha_{2}(z)}$, where $Q_{d_{*}}(z,f)$ and $Q_{d}(z,f)$ are differential polynomials in $f$ with small functions as coefficients, of degree $d_{*}\, (\leq n-1)$ and $d\, (\leq n-2)$ respectively, $R,\, p_{1},\, p_{2}$ are non-vanishing small functions of $f$, and $\alpha,\, \alpha_{1},\, \alpha_{2}$ are nonconstant entire functions. In particular, we give out the conditions for ensuring the existence of these kinds of meromorphic solutions and their possible forms of the above equations.
Keywords : Meromorphic functions; nonlinear differential equations; small functions; differential polynomials.
MSC numbers : 34M05; 30D30; 30D35
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