wave equation(doppler effect)

Solve the equation of wave equation with fixed source at the origin $$\Delta p + \omega^2p=-\delta(x)$$ The question asks for solution of the form $$p(r) = C\frac{e^{i\lambda r}}{r}$$, where $$r = |x|$$.

I first plug the proposed solution $$p$$ into the equation when $$x$$ is away from 0(so that the equation becomes homogeneous) and get that $$\lambda = \pm\omega$$. Then I integrate the equation over a small ball around 0. $$\int_{\partial B_\epsilon(0)}\frac{\partial p}{\partial r}dS=\int_{B_\epsilon(0)}\Delta pdx = \int_{B_\epsilon(0)}(\delta(x) - \omega^2p) dx = 1-\omega^2\int_{B_\epsilon(0)}pdx$$

I follow the hints and get the above results, but I don't know how to deal with the first and last term in the last equation that I obtain to finally determine the $$\lambda$$ and $$C$$. Can someone give some ideas?

• I think the first term should be $\int_{\partial B_{\epsilon}(0)}\frac{\partial p}{\partial r}dS = 4\pi \epsilon^2 p_r(\epsilon)$, right? – QD666 Oct 22 '18 at 0:32