I am wandering how to solve non-homogeneus screened Poisson equaation: $$ \nabla^2 \phi - \lambda^2 \phi=f(\mathbf{r}) $$ My function $f(\mathbf{r})$ is a constant all over the place, i denote it's velue with D, therefore $\nabla^2 \phi - \lambda^2 \phi=D$. I was able to solve this for symetric solutions in spherical coordinates with solution $\phi=a \frac{e^{-\lambda r}}{r}+b\frac{e^{-\lambda r}}{r} $. Furthermore I have boundry contition that potential $phi$ doesn't diverge at $r=\infty$.
But I don't know how to approach for non-homogenus solution. According to Wikipedia one can use Green's function approach with Green's function for this case: $$ G=\frac{e^{-\lambda r}}{4 \pi r}, $$ but I have some problems with integration: $$ \phi=\int d^3 r' G(r-r')f(r')=F\int_0 ^{\infty}r'^2\frac{e^{-\lambda |r-r'|}}{|r-r'|}dr $$ But the following integal does not converge, according to all my tries. Is there any other more suitable method for Solving upper equation? Or is my Greeen's formula approach wrong or my version of Green's formula?