On the growth of solutions of some higher order linear differential equations with meromorphic coefficients

Аннотация

In this paper, by using the value distribution theory, we study the growth and the oscillation of meromorphic solutions of the linear differential equation \begin{align*} f^{(k) }&+\left( A_{k-1,1}(z) e^{P_{k-1}(z) }+A_{k-1,2}(z) e^{Q_{k-1}(z) }\right) f^{\left( k-1\right) } \\ & +\cdots +\left( A_{0,1}(z) e^{P_{0}(z) }+A_{0,2}(z) e^{Q_{0}(z) }\right) f=F(z), \end{align*} where $A_{j,i}(z) \left( \not\equiv 0\right) $ $\left( j=0,\ldots,k-1\right),$ $F(z) $ are meromorphic functions of a finite order, and $P_{j}(z),Q_{j}(z) $ $ (j=0,1,\ldots,k-1;i=1,2)$ are polynomials with degree $n\geqslant 1$. Under some conditions, we prove that as $F\equiv 0$, each meromorphic solution $f\not\equiv 0$ with poles of uniformly bounded multiplicity is of infinite order and satisfies $\rho _{2}(f)=n$ and as $F\not\equiv 0$, there exists at most one exceptional solution $f_{0}$ of a finite order, and all other transcendental meromorphic solutions $f$ with poles of uniformly bounded multiplicities satisfy ${\overline{\lambda }(f)=\lambda (f)=\rho \left( f\right) =+\infty }$ and $\overline{\lambda }_{2}\left( f\right) =\lambda _{2}\left( f\right) =\rho _{2}\left( f\right) \leq \max \left\{ n,\rho \left( F\right) \right\}.$ Our results extend the previous results due Zhan and Xiao [19].