# Can't verify solution to Debye-Huckel equation

In the derivation of Debye length (https://en.wikipedia.org/wiki/Debye_length), the Debye-Huckel equation for a neutral system is

$$\nabla^2 \Phi(r) = \frac{\Phi(r)}{\lambda_D^2} - \frac{Q \delta^3(\mathbf{r})}{\varepsilon}$$.

The solution is given as

$$\Phi(r) = \frac{Q}{4 \pi \varepsilon r} e^{-r / \lambda_D}$$.

However, I am unable to verify that this solution satisfies the equation. Here is my work.

$$\nabla^2 \Phi(r) = \frac{Q}{4 \pi \varepsilon}e^{-r / \lambda_D} \nabla^2 \frac{1}{r} + \frac{Q}{4 \pi \varepsilon r} \nabla^2 e^{-r / \lambda_D}$$

$$\nabla^2 \Phi(r) = -\frac{Q\delta^3(\mathbf{r})}{\varepsilon}e^{-r / \lambda_D} + \frac{Q}{4 \pi \varepsilon r} \left( -\frac{2}{r \lambda_D} e^{-r/\lambda_D} + \frac{1}{\lambda_D^2} e^{-r / \lambda_D} \right)$$

$$\nabla^2 \Phi(r) = -\frac{Q\delta^3(\mathbf{r})}{\varepsilon}e^{-r / \lambda_D} - \frac{2Q}{4 \pi \varepsilon r^2 \lambda_D}e^{-r / \lambda_D} + \frac{\Phi(r)}{\lambda_D^2}$$

Where is my mistake?

You are missing the cross term, proportional to $$\left(\vec{\nabla}\frac{1}{r}\right)\cdot\left[\vec{\nabla}\exp(r/\lambda_{D})\right]$$, in the first line of your calculation.