Elliptic Weingarten surfaces of minimal type in $\mathbb{R}{\times }_h\mathbb{R}$

In this paper, we study the elliptic Weingarten surfaces of minimal type immersed in the warped product space $\mathbb{R}{\times }_h\mathbb{R}$, when $h$ is a $C^1$-function in ${\mathbb{R}}^2$ with radial symmetry. That is, surfaces whose mean curvature $H$ and extrinsic curvature $K$ satisfy a relationship $H=f(H^2-K)$ where $f\in C^1(-\epsilon ,+\mathrm{\infty })$ with $\epsilon >0$, $f(0)=0$ and $4t(f^{\mathrm{\prime }}(t){)}^2<1$ for $t\in (-\epsilon ,\mathrm{\infty })$. We show, under some assumptions about the warping function $h$, the existence and uniqueness of the rotationally-invariant examples of elliptic Weingarten of minimal type surfaces immersed in $\mathbb{R}{\times }_h\mathbb{R}$ as well as we study the geometric behavior of its generating curve.