Considering a family of rational map $U_{mn\lambda }$ of the renormalization transformation of the generalized diamond hierarchical Potts model, we give the asymptotic formula of the Hausdorff dimension of the Julia sets of $U_{mn\lambda }$ as the parameter $\lambda$ tends to infinity, here

$$U_{mn\lambda }={\left[\frac{{\left(z+\lambda -1\right)}^n+\left(\lambda -1\right){\left(z-1\right)}^n}{{\left(z+\lambda -1\right)}^n-{\left(z-1\right)}^n}\right]}^m,$$

where $m\ge 2$, $n\ge 2$ are two natural numbers, $\lambda \in \mathbb{C}$.

By using Nevanlinna of the value distribution of meromorphic functions, we investigate the transcendental meromorphic solutions of the non-linear differential equation

$$f^n+P_d(f)=p_1{e}^{{\alpha }_{1}z}+p_2{e}^{{\alpha }_{2}z}+p_3{e}^{{\alpha }_{3}z},$$

where $P_d(f)$ is a differential polynomial in $f$ of degree $d(0\le d\le n-3)$ with small meromorphic coefficients and $p_i,{\alpha }_i(i=1,2,3)$ are nonzero constants. We show that the solutions of this type equation are exponential sums and they are in ${\mathrm{\Gamma }}_0\cup {\mathrm{\Gamma }}_1\cup {\mathrm{\Gamma }}_3$ which will be given in Section $1$. Moreover, we give some examples to illustrate our results.