How to solve a PDE with a Dirac Delta and what does the PDE means? If I have a PDE $ \Delta u= \delta(0)$ on some bounded domain in $\mathbb{R}^2$ with smooth boundary with some nice enough boundary condition.
What is the solution of the PDE? And what is the PDE mean in term of integral?
Because I know  $\delta(0)$ is not a function.
Thank you

This is a symbolic statement of the fact that the $u$ in question is the fundamental solution/Greens function of the Laplace operator on the diagonal. For a discussion of the concept have a look at this wikipedia article.
