Solve the screened Poisson equation

1. The problem statement, all variables and given/known data

Solve the equation

[tex]

\nabla^2\phi-\frac{1}{\lambda^2_D}\phi=-\frac{q_T}{\epsilon_0}\delta(r)

[/tex]

substituting the [itex]\delta[/itex] representation

[tex]

\delta(r)=\frac{1}{4\pi}\frac{q_T}{r}

[/tex]

and writing the laplacian in spherical coordinates. Use as your guess

[tex]

\phi=\frac{1}{4\pi\epsilon_0}\frac{g(r)}{r}

[/tex]

and show

[tex]

g(r)=\exp(-r/\lambda_D)

[/tex]



3. The attempt at a solution

I know how to solve this equation using the Green and Fourier formalism, but here I am asked to solve it "the easy way" (as said in "Introduction to plasma physics - Bellan").



All I can say is

[tex]

\nabla^2\left(\phi-\frac{q_T}{4\pi\epsilon_0}\frac{1}{r}\right)-\frac{\phi}{\lambda^2}=0

[/tex]

Using the guess

[tex]

\phi=\frac{1}{4\pi\epsilon_0}\frac{g(r)}{r}

[/tex]

I have

[tex]

\frac{1}{r^2}\partial_r\left[r^2\partial_r\left(\frac{g(r)-q_T}{4\pi\epsilon_0r}\right)\right]-\frac{g}{4\pi\epsilon_0r\lambda^2}=0

[/tex]

[tex]

\frac{1}{r^2}\partial_r\left[r^2\left(\frac{r\partial_r g-(g-q_T)}{r^2}\right)\right]-\frac{g}{r\lambda^2}=0

[/tex]

[tex]

\frac{1}{r^2}(\partial_r g+r\partial_r^2g-\partial_r g)-\frac{g}{r\lambda^2}=0

[/tex]

And so the final differential equation for g(r) is

[tex]

\frac{\partial_r^2g}{r}-\frac{g}{r\lambda^2}=0

[/tex]

which has the physical solution

[tex]

g(r)=\exp(-r/\lambda)

[/tex]



This result is correct, but my doubt is: the [itex]\delta[/itex] term didn't play any role, I mean, if I solved the equation

[tex]

\nabla^2\phi-\frac{1}{\lambda^2_D}\phi=0

[/tex]

I would have obtained the same result.

I suppose that without the [itex]\delta[/itex] term there is some problem in r=0, but I don't fully understand what is the problem