  We study the existence, uniqueness and minimality of critical points of the
form $m_{\varepsilon,\eta}(x) = (f_{\varepsilon,\eta}(|x|)\frac{x}{|x|},
g_{\varepsilon,\eta}(|x|))$ of the functional \[ E_{\varepsilon,\eta}[m] =
\int_{B^N} \Big[\frac{1}{2} |\nabla m|^2 + \frac{1}{2\varepsilon^2} (1 -
|m|^2)^2 + \frac{1}{2\eta^2} m_{N+1}^2\Big]\,dx \] for $m=(m_1, \dots, m_N,
m_{N+1}) \in H^1(B^N,\mathbb{R}^{N+1})$ with $m(x) = (x,0)$ on $\partial B^N$.
We establish a necessary and sufficient condition on the dimension $N$ and the
parameters $\varepsilon$ and $\eta$ for the existence of an escaping vortex
solution $(f_{\varepsilon,\eta}, g_{\varepsilon,\eta})$ with
$g_{\varepsilon,\eta}> 0$. We also establish its uniqueness and local
minimality. In the limiting case $\eta = 0$, we prove the local minimality of
the degree-one vortex solution for the Ginzburg-Landau (GL) energy for every
$\varepsilon > 0$ and $N \geq 2$. Similarly, when $\varepsilon = 0$, we prove
the local minimality of the degree-one escaping vortex solution to an
$\mathbb{S}^N$-valued GL model arising in micromagnetics for every $\eta > 0$
and $2 \leq N \leq 6$.
