Meromorphic solutions of a nonlinear differential-difference equation with two exponential terms

Abstract

Let $n\geq 2$ and $k\geq 1$ be integers. We study finite order meromorphic solutions of the nonlinear differential-difference equation \begin{equation*} f^{n}(z)f'(z)+P_{d}(z,f)=p_{1}(z){\rm e}^{α_{1}z^{k}}+p_{2}(z){\rm e}^{α_{2}z^{k}}, \end{equation*} where $P_{d}(z,f)\not\equiv 0$ is a difference polynomial in $f$ of degree $d\leq n-1$ whose coefficients are small meromorphic functions with respect to $f$, $p_{1},~p_{2}$ are nonzero meromorphic functions with $T(r,p_{j})=o(r^{k})$, $j=1,2$, and $α_{1}\neqα_{2}$ are nonzero constants. We prove that every such solution has order exactly $k$, and obtain sufficient conditions on $n,~d,~k$ for the equation to admit no finite order meromorphic solution with $N(r,\frac{1}{f})=S(r,f)$. Our results extend the known ones from linear exponents $z$ to polynomial exponents $z^{k}$, and replace the linear difference polynomials with the degree restriction $d\leq n-2$ assumed in the literature by general ones of degree $d\leq n-1$. Examples show that the restrictions on $n$ and $d$ are sharp.