I'm trying to solve a system of linear BVPs and would appreciate any help or pointers.
The system is given by
$$ - \Delta n(x) + \frac{d^{2}}{d x^{2}} \Delta \psi(x) = 0 \tag{1}\label{1} $$
and
$$ -j_0 - \Delta n(x) \frac{d}{d x}\psi(x)-n(x) \frac{d}{d x}\Delta \psi(x) + \frac{d}{dx} \Delta n(x) = 0.\tag{2}\label{2} $$
The boundary conditions are $\Delta n(0) = \Delta n(1) = \Delta \psi(0) = \Delta \psi(1)=0$. We can also assume $n(x) > 0$ for $x \in [0,1]$.
The question is: Under what conditions on $n(x)$, $\psi(x)$ is this system solvable with the given boundary conditions?
We can, in principle, find a solution by solving $\ref{2}$ for $\Delta \psi(x)$, taking the derivative w.r.t. $x$ and substituting it into $\ref{1}$. This gives us a second-order equation for $\Delta n(x)$. We can write it in self-adjoint form as
$$ -\frac{d}{dx} \left[\frac{e^{-\psi(x)}}{n(x)} \frac{d}{dx} \Delta n(x)\right] - \frac{e^{-\psi(x)}}{n(x)} \left[ -n(x) - \frac{d^2}{d x^2}\psi(x) + \frac{1}{n(x)} \frac{d}{dx} \psi(x) \frac{d}{dx} n(x) \right] \Delta n(x)= j_0 \frac{e^{-\psi(x)}}{n^2(x)} \frac{d}{dx} n(x). \tag{3}\label{3} $$
Given a solution for $\ref{3}$, we can find $\Delta \psi(x)$ by integrating $\ref{2}$:
$$ \Delta \psi(x) = C_0 + \int_0^x \frac{1}{n(\xi)} \left[-j_0 - \Delta n(\xi) \frac{d}{dx} \psi(x) + \frac{d}{dx} \Delta n(x)\right]\,d\xi. $$
The boundary condition $\Delta \psi(0) = 0$ requires $C_0=0$. The boundary condition $\Delta \psi(1)=0$ then translates into an integral condition
$$ \int_0^1 \frac{1}{n(\xi)} \left[-j_0 - \Delta n(\xi) \frac{d}{dx} \psi(x) + \frac{d}{dx} \Delta n(x)\right]\,d\xi = 0. $$
This condition seems rather restrictive as we don't have any integration constants left. I believe that $j_0$ does not help us here since we can always scale $\Delta \psi(x)$ and $\Delta n(x)$ simultaneously by $1/j_0$ and then get a system which is independent of $j_0$.
Now, are there any $n(x)$, $\psi(x)$ for which this system is solvable? Are there any statements we can make about $n(x)$, $\psi(x)$ for this system to be solvable?
Also I'd be interested in any pointers to the theory of systems of linear BVPs. All books I know only cover single BVPs.
Thanks for reading so far. Any help would be much appreciated.