Local minimality of $\mathbb{R}^N$-valued and $\mathbb{S}^N$-valued Ginzburg–Landau vortex solutions in the unit ball $B^N$

We study the existence, uniqueness and minimality of critical points form $m_{\varepsilon,\eta}(x) = (f_{\varepsilon,\eta}(|x|)\frac{x}{|x|}, g_{\varepsilon,\eta}(|x|))$ 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 $\partial B^N$. establish a necessary sufficient condition dimension $N$ parameters $\varepsilon$ $\eta$ existence an escaping vortex solution $(f_{\varepsilon,\eta}, g_{\varepsilon,\eta})$ $g_{\varepsilon,\eta}> 0$. also its local minimality. In limiting case $\eta 0$, we prove degree-one Ginzburg-Landau (GL) energy every $\varepsilon > 0$ $N \geq 2$. Similarly, when to $\mathbb{S}^N$-valued GL model arising in micromagnetics $2 \leq N 6$.