Not sure what is wrong with the boundary conditions for this problem:
NDSolve[{D[u[x, y], x, x] + D[u[x, y], y, y] -
u[x, y] == -DiracDelta[x, y], u[10, y] == 0, u[x, 10] == 0,
u[-10, y] == 0}, u, {x, 0, 10}, {y, 0, 10}].
The error message is:
NDSolve::ivone: Boundary values may only be specified for one independent variable. Initial values may only be specified at one value of the other independent variable.
>>
after the error message. The other problem (which is not the cause of the error) is that you can't (it doesn't make sense to) use DiracDelta in a numerical method. $\endgroup$