  In this article, we firstly study the cone Moser-Trudinger inequalities and
their best exponents $\alpha_2$ on both bounded and unbounded domains
$\mathbb{R}^2_{+}$. Then, using the cone Moser-Trudinger inequalities, we study
the existence of weak solutions to the nonlinear equation \begin{equation*}
\left\{\begin{array}{ll} -\Delta_{\mathbb{B}} u=f(x, u), &\mbox{in}\ x\in
\mbox{int} (\mathbb{B}), \\ u= 0, &\mbox{on}\ \partial\mathbb{B}, \end{array}
\right. \end{equation*} where $\Delta_{\mathbb{B}}$ is Fuchsian type Laplace
operator investigated with totally characteristic degeneracy on the boundary
$x_1 =0$, and the nonlinearity $f$ has the subcritical exponential growth or
the critical exponential growth.
