  In this paper, we study the transcendental meromorphic solutions for the
nonlinear differential equations: $f^{n}+P(f)=R(z)e^{\alpha(z)}$ and
$f^{n}+P_{*}(f)=p_{1}(z)e^{\alpha_{1}(z)}+p_{2}(z)e^{\alpha_{2}(z)}$ in the
complex plane, where $P(f)$ and $P_{*}(f)$ are differential polynomials in $f$
of degree $n-1$ with coefficients being small functions and rational functions
respectively, $R$ is a non-vanishing small function of $f$, $\alpha$ is a
nonconstant entire function, $p_{1}, p_{2}$ are non-vanishing rational
functions, and $\alpha_{1}, \alpha_{2}$ are nonconstant polynomials.
Particularly, we consider the solutions of the second equation when $p_{1},
p_{2}$ are nonzero constants, and $\deg \alpha_{1}=\deg \alpha_{2}=1$. Our
results are improvements and complements of Liao (Complex Var. Elliptic Equ.
2015, 60(6): 748--756), and Rong-Xu (Mathematics 2019, 7, 539), etc., which
partially answer a question proposed by Li (J. Math. Anal. Appl. 2011, 375:
310--319).
