On the solvability of a semilinear higher-order elliptic problem for the vector field method in image registration

We study the existence of the solution to a semilinear higher-order elliptic system

\begin{document}$ \mathcal{L}v(t, \cdot) = F_{S, T}\circ G(v)(t, \cdot), \quad \forall t\in [0, \tau], $\end{document}

with the homogeneous Dirichlet boundary conditions. Here, \begin{document}$ \mathcal{L} = (-\Delta)^m $\end{document} is a harmonic operator of order \begin{document}$ m $\end{document}, \begin{document}$ v = v(t, x):[0, \tau]\times\Omega\rightarrow \mathbb{R}^n $\end{document} is the unknown, \begin{document}$ t $\end{document} is a parameter, \begin{document}$ F_{S, T} $\end{document} is a function related to given functions \begin{document}$ S $\end{document} and \begin{document}$ T $\end{document}, and \begin{document}$ G(v)(t, x) $\end{document} is defined by the solution \begin{document}$ y^v(s;t, x) $\end{document} of an ODE-IVP \begin{document}$ {\rm d}y/\mathrm{d}s = v(s, y), \quad y(t) = x $\end{document}. The elliptic equations is the Euler-Lagrange equation of the vector field regularization model widely used in image registration. Although we have showed the existence of a solution to this BVP by the variational method, we hope to study it by the fixed point method further. This is mainly because the elliptic equations is novel in form, and the method here put more emphasis on the quantitative analysis whereas the variational method focus on the qualitative analysis. Since the system here is a higher order semilinear system, and its nonlinear term is dominated by an exponential function with respect to the unknown, we use an exponential inequality to construct a closed ball, and then apply the Schauder fixed point theorem to show the existence of a solution under some assumptions.