Maple 2018 solves the ODE under consideration:
dsolve(((D@@2)(w))(r)+(D(w))(r)/r-k^2*w(r) = -g(r))
$$w \left( r \right) ={{\ I}_{0}\left(kr\right)}{\it \_C2}+{{ K}_{0 }\left(kr\right)}{\it \_C1}-\int \!{{ K}_{0}\left(kr\right)}g \left( r \right) r\,{\rm d}r{{ I}_{0}\left(kr\right)}+$$ $$\int \!{ { I}_{0}\left(kr\right)}g \left( r \right) r\,{\rm d}r{{ K}_{0} \left(kr\right)} $$
The assumption $w(r)$ decays exponentially fast implies the constants ${\it \_C1}$ and ${\it \_C2} $ equal zero because of
asympt(BesselI(0, k*r), r, 2)
$$\frac 1 2\,{\frac {\sqrt {2}\sqrt {{k}^{-1}}{{\rm e}^{kr}}\sqrt {{r}^{-1}}}{ \sqrt {\pi}}}+O \left( \left( {r}^{-1} \right) ^{3/2} \right) $$
Hope this would be useful.