On meromorphic solutions of certain nonlinear differential equations. (English) Zbl 1047.34101

The authors deal with differential equations of the form \[ L(f)+ p(z, f)= h(z),\tag{1} \] where \(L(f)\) is a linear differential polynomial in \(f\) with meromorphic coefficients, \(p(z, f)\) is a polynomial in \(f\) with meromorphic coefficients, and \(h(z)\) is meromorphic. Define \(L_f:= \{h\) meromorphic: \(T(r,h)= S(r,h)\}\) and denote by \(F\) the family of meromorphic solutions to (1) such that, whenever \(f\in F\), all coefficients in (1) are in \(L_f\), and \(N(r\cdot f)= S(r\cdot f)\). It follows that, if \(f,g\in F\), then \[ T(r\cdot g)= O(T(r\cdot f))+ S(r\cdot f). \] Moreover, if \(\alpha> 1\), then, for some \(r_\alpha> 0\), \[ T(r\cdot g)= O(T(\alpha r,f)) \] for all \(r\geq r_\alpha\). The authors show that, if \(f\) is a meromorphic solution to (1) such that all coefficients in (1) are in \(L_f\), then \(\rho(f)\geq \rho(h)\). If \(n=: \deg_f p(z, f)\geq k+2\) and \(N(r\cdot f)= S(r\cdot f)\) then \(\rho(f)= \rho(f)\) and \(\mu(f)= \mu(f)\).
For the equation \[ L(f)- p(z) f^n= h(z), \] \(h(z)\) be a meromorphic function, the authors show that the method used by Yang can be modified to obtain similar uniqueness results for meromorphic solutions to this generalized equation, when \(n\geq 4\).


34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
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