Three results on transcendental meromorphic solutions of certain nonlinear differential equations

In this paper, we study the transcendental meromorphic solutions for the nonlinear differential equations: $f^n+P(f)=R(z)e^{\alpha (z)}$ and $f^n+P_{\ast }(f)=p_1(z)e^{{\alpha }_1(z)}+p_2(z)e^{{\alpha }_2(z)}$ in the complex plane, where $P(f)$ and $P_{\ast }(f)$ 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 $p_{1}, p_{2}$arenonzeroconstants,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).