New Harnack inequalities and existence theorems for radially symmetric solutions of elliptic PDEs with sign changing or vanishing Green's function

We consider the perturbed Hammerstein integral equationy(t)=γ1(t)H1(φ1(y))+γ2(t)H2(φ2(y))+λ∫01G(t,s)f(s,y(s))ds, where φ1 and φ2 are linear functionals realized as Stieltjes integrals with associated signed Stieltjes measures. We demonstrate that by utilizing a nonstandard order cone together with an associated nonstandard open set, existence of a positive solution to the integral equation can be guaranteed under relatively mild hypotheses on the various constituent functions and functionals in the case in which the Green's function (t,s)↦G(t,s) is allowed to both vanish and change sign. As an example we apply our results to radially symmetric elliptic PDEs of the form−Δu=λh(|x|)g(u(x)), |x|∈[R1,R2]∂∂ru(x)|x∈∂BR2=0(u(R1x)+u(Rηx))|x∈∂B1=H(φ(u)), where 0