I am reading a paper in which it is proposed that one can solve a problem from mathematical physics by establishing an existence theory for a system of equations. One of the equations in the system is not a PDE and so the system is not local: it's not written in the paper, but it should be an integral of $t$ taken from $0$ to $\infty$ (hence it does not satisfy a local PDE at every point). To clarify, the other two equations in the system are elliptic PDEs.
In order to have an existence theory for the system, one would have to show that the theory works when $\phi$ takes a 'wide range of possible values', but I find it unclear what it would mean for $\phi$ to take a sufficiently wide range of values.
Edit: System of equations added (first equation is the Dirac equation). $$ \begin{cases} D \tilde{\psi_t} = 0,\\ \phi = 2 \displaystyle\int\limits_0^{\infty}\! \chi_t \bigg((w_t^{-})^2|\psi_t^{-}|^2 + (w_t^{+})^2|\psi_t^{+}|^2 \bigg) u_t^2 \: \text{d} t \\ H_{\Sigma} - Tr_{\Sigma} K = 0 \\ \end{cases} $$ It has been requested that I define all the symbols being used: this is a bit lengthy but geometrically interesting so here goes. Start with a Riemannian $3$-manifold equipped with a metric $g$ and a symmetric $2$-tensor field $k$. We will now define the metric given by conformal flow of metrics: this is used in papers by Bray and Agol, Storm and Thurston. This flow is given by
$$ \begin{cases} g_t = u_t^4 g,\\ \frac{d}{dt} u_t = v_t u_t\\ \Delta_{g_t} v_t =0 \:\: \text{on} \: M_t \\ \end{cases} $$ where $v_t =0$ on $\partial M_t$ and $v_t(x)$ tends to $-1$ as $|x|$ tends to $\infty$. Assuming $(M,g_t)$ contains an outermost minimal surface, $\partial M_t$ is the region outside of that surface. Take two copies of $M_t$ denoted by $M_t^-$ and $M_t^+$, where the copy is equipped with a metric $g_t^+ = (w_t^+)^4 g_t$ and $w_t^+ = (1 + v_t)/2$ (swap plus with minus for the other metric).
Now take what Bray refers to as the doubled manifold $\tilde{M}_t$, which is the union of $M_t^-$ and $M_t^+$. The associated metric $\tilde{g}_t$ is defined piecewise as the union of $g_t^-$ and $g_t^+$. The spinors in the equations are defined on the manifolds you would expect.
$\Sigma$ is a hypersurface which is given by a graph $t=f(x)$ which sits inside a general warped product space equipped with a warped product metric $(M \times \mathbf{R}, g + \phi^2 dt^2)$, for some function $\phi$ defined on $M$. $\chi_t(x)$ is defined to be $1$ when $x$ is in $\Sigma_t$ and $0$ when $x$ is in $\Sigma_0 - \Sigma_t$. The notation denotes that the conformal flow is now being performed on $\Sigma_0$. The third equation can be written in local coordinates to see the dependence on $g$, $k$ and the warping factor $\phi$, but essentially it is a second-order quasi-linear elliptic PDE. $K$ is just the extension of $k$ on $M$ to the product space and $H$ is the mean curvature.
So this is the issue really, what kind of existence theory will one have for the system to deal with the second integral equation. The coupling between $g$ and $\phi$ is highly non-trivial.
