Estimates for the green function and existence of positive solutions for higher-order elliptic equations

We establish a 3G-theorem for the iterated Green function of (−∆)pm, (p≥1,m≥1), on the unit ball B of ℝn(n≥1) with boundary conditions (∂/∂ν)j(−∆)kmu=0 on ∂B, for 0≤k≤p−1 and 0≤j≤m−1. We exploit thisresult to study a class of potentials m,n(p). Next, we aim at proving the existence of positive continuoussolutions for the following polyharmonic nonlinear problems (−∆)pmu=h(‧,u), in D (in the sense of distributions), lim|x|→1((−∆)kmu(x)/(1−|x|)m−1)=0, for 0≤k≤p−1, where D=B or B\{0} and h is a Borel measurablefunction on D×(0,∞) satisfying some appropriateconditions related to m,n(p)