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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 \(n\geq 4\).

MSC:

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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References:

[1] DOI: 10.1007/BF02785417 · Zbl 1016.34091
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[6] DOI: 10.1016/0022-247X(85)90216-1 · Zbl 0593.34014
[7] DOI: 10.1007/BF02807430 · Zbl 0129.29301
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