In some textbooks I have seen the Green's equation in this way: $$ L G(x,s) = \delta(x - s)$$ But in some other. $$ L G(x,s) = -\delta(x - s)$$
In Wikipedia, they say the change in sign is used in physical problems, but don't give a reason for this. May some one give me a reason about it? Thanks
In some problems in physics, the operator $\mathscr{L}$ operates on a object to yield the negative of another object.
For example, in electrostatics, we have $\mathscr{L}=\nabla^2$ that operates on the electrostatic potential $\Phi(\vec r)$ and yields the negative of the volume charge density $\rho(\vec r)$ divided by the permittivity of free space. That is to say,
$$\nabla^2 \Phi(\vec r)=-\frac{\rho(\vec r)}{\epsilon} \tag 1$$
with suitable boundary conditions given to provide a well-posed problem (e.g., Dirichlet or Neumann boundary conditions).
Then, if the Green's Function $G(\vec r|\vec r')$ is the solution of
$$\nabla^2 G(\vec r|\vec r')=-\delta(\vec r-\vec r')$$
then the solution to $(1)$ for the free-space problem (i.e., unbounded problem) is
$$\Phi(\vec r)=\frac{1}{\epsilon}\int_{R^3}\rho(\vec r')\,G(\vec r|\vec r')\,dV' \tag 2$$
which avoids the use of the minus sign on the right-hand side of $(2)$.
And that is about all there is to the rationale for the convention.