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 $f \in C^{1}(-\epsilon,+\infty)$ with $\epsilon > 0$, $f(0)=0$ and $4t(f'(t))^{2} < 1$ for $t \in (-\epsilon,\infty)$. 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.