In a mathematical physical problem, I came across the following partial differential equation involving a delta Dirac function: $$ a \, \frac{\partial^2 w}{\partial x^2} + b \, \frac{\partial^2 w}{\partial y^2} + \delta^2(x,y) = 0 \, , $$ subject to the boundary conditions $w(x = \pm 1, y) = w(x, y = \pm 1) = 0$. Here $a, b \in \mathbb{R}_+$ and $\delta^2(x,y) = \delta(x)\delta(y)$ is the two-dimensional delta Dirac function.
While solutions for ODEs with delta Dirac functions can readily be obtained using the standard approach, I am not aware of any resolution recipe for PDEs with delta Dirac functions.
Any help or hint is highly desirable and appreciated.
Thank you