# Non-linearities in Lagrangian of a scalar field coupled to point-like source

I have an exercise where I did not manage to understand the questions. Basically, I have this Lagrangian

$$\mathcal{L}=\frac{1}{2}(\partial \pi)^2-\frac{1}{\Lambda^3}(\partial \pi)^2\square \pi +\alpha \pi \delta^3(\vec{x})$$

The questions are:

1. Discuss at what distance from the source the non-linearities of the $\pi$ field become important in the spherically symmetric solution.
2. Compute the field $E(r)$ generated by the point like source in the spherically symmetric soultion, where $\vec{E}=\vec{\nabla} \pi=\hat{r}E(r)$

What does question $1$ mean? I need to solve the equation of motion in order to find the spherically symmetric solution and then maybe take some limit for $r\rightarrow 0$, how can I solve the equation of motion?

Question $2$ maybe is not difficult if I manage to understand question $1$, so I would like to ask something else: when I have a Lagrangian of a particle coupled to a gauge field, how can I "define" an electromagnetism? I mean, how I find the electric and magnetic field generated by the particle inside an external field, in this case represented by a point-like source?

In both cases I will be happy if just someone can tell me some books or notes where I can find the answers to both my questions.

$$\frac{ (\partial \pi)^2 \Box \pi }{\Lambda^3} \sim \alpha\, \pi\, \delta(\vec{x})$$
1. Here you are being asked to solve the e.o.m, in the case where $\pi =\pi(r)$. Indeed, using spherical coordinates for the metric, once you plug $r E(r)= \partial_r \pi$ into the e.o.m and solve it you will end up obtaining a fairly simple expression for $E$.