# Reference request for weak solutions of an Elliptic PDE

Edit : I just learned that all weak solutions are $C^\infty$, so this question, by Willie, seems more appropriate than the current one.

I want to find weak, non trivial, continuous, solutions of $$\Delta u - \lambda u = 0$$ for a square domain in $\mathbb{R}^N$, $N \ge 2$, under periodic boundary conditions, and under an added constraint that, the weak solutions $u$ should take given values, at a given finite set of points in the interior of the domain. $u(x_i) = d_i$, $x_i$ lie in the interior of the domain, and $d_i$ are reals.

Reference request, if someone already solved it, or partially solved it or any relevant work. I am trying to solve it and I want to know if it makes sense, and I am not re-inventing, or barking up the wrong tree.

PS : Solving, I mean, having a numerical solution that converges pointwise, to the actual solution.

Firstly, keeping in mind the well-posedness of Elliptic PDE, I would start working with the case of $\lambda \geq 0$. And to make things easier have a Dirichlet boundary, Periodic boundary wouldn't be very different, we will the have to consider the point on the boundary of the given space to also be active nodes, but there is a subtle problem here, how do we define connectivities of these nodes with the other boundary nodes? By the taking the nodes from the internal of the other boundary being place at the same spacing on the empty side of the other periodic boundary, like ghost points and then making the connectivities.