Some Studies On Meromorphic Solutions Of Nonlinear Differential Equations

In this thesis,we use the value distribution theory of meromorphic functions as a main tool.We analyze the solutions of nonlinear complex differential equation of the form fⁿ+P_d（f）=p₁e^{α₁（z）}+p₂e^{α₂（z）}.This is a hot research at home and abroad in recent years.Thesis is divided into four chapters,and the concrete arrangement is as follows:In chapter 1,we briefly introduce Nevanlinna value distribution theory,WimanValiron theory and difference analogues of meromorphic functions.In chapter 2,we mainly discuss the differential equation f⁴+a（z）ff’=p₁（z）e^{α₁（z）}+p₂（z）e^{α₂（z）}studied by Zhang in 2018.We consider the structure of the solutions of differential equation f⁴+a（z）ff^{（k）}=p₁（z）e^{α₁（z）}+p₂（z）e^{α₂（z）}when f’ is replaced by f^{（k）}.We obtain the concrete form of the solutions of the above equation,and present that the quadratic differential monomial a（z）ff^{（k）}in the above equation cannot be replaced by quadratic differential polynomial.In addition,we analyze the existence of meromorphic solutions of another type of nonlinear complex differential equations f³+a（z）f’=p₁（z）e^{ν（z）}+p₂（z）e^{-ν（z）}.The results obtained generalize the relevant conclusions of Zhang.In chapter 3,we consider the question raised by Li in 2011.How to find the solutions of equation fⁿ+P_d（f）=p₁e^α₁z+p₂e^α₂z under the condition deg P_d（f）=n-1?Inspired by the study of Li and Yang,we consider the form of the solutions of the equation when every exponential part on the right side of the equations is the kth order monomial.We delete the condition that f is a transcendental meromorphic function and obtain the form of the solutions of the equation.Moreover,we give a number of examples to present the necessity of the condition N（r,f）=S（r,f）.These results extend the results of Li and Yang.In chapter 4,we make a summary and pose some problems.