Diagonals of Green’s functions and applications

We describe the diagonal of the Green’s Function for some Sturm–Liouville operators, as it was introduced and studied in a number of papers already, and we discuss a number of applications that depend on its properties. Semilinear elliptic PDE’s with rotationally invariant and (possibly) singular coefficients, with power type nonlinearities, provide a particularly well suited class of examples where the results of this article apply.