I want to find a distribution $E$ which satisfies $(I - \Delta) E = \delta$.
I tried the usual method of finding the fundamental solution of Laplace operator. Using polar coordinate and calculating $ \langle (I-\Delta)E,\phi \rangle = \langle E,(I-\Delta)\phi \rangle$
$$\langle E, (I-\Delta) \phi \rangle = 2 \pi \int_{0}^{\infty} E(r) \left(r-\frac{d^{2}}{dr^{2}} - \frac{1}{r}\frac{d}{dr} \right)\overline{\phi} dr$$
Here
$$\overline{\phi}(r)=\frac{1}{2\pi}\int_{0}^{2\pi}\phi(r,\theta)d\theta$$
In the usual approach to finding the fundamental solution of the Laplace operator, we integrate by parts. But I cannot integrate by parts in this case.
Any help will be appreciated.