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

Abstract

We establish a3G-theorem for the iterated Green function of(−∆)pm, (p≥1,m≥1), on the unit ballBofℝn(n≥1)with boundary conditions(∂/∂ν)j(−∆)kmu=0on∂B, for0≤k≤p−1and0≤j≤m−1. We exploit this result to study a class of potentials𝒥m,n(p). Next, we aim at proving the existence of positive continuous solutions for the following polyharmonic nonlinear problems(−∆)pmu=h(‧,u), inD(in the sense of distributions),lim|x|→1((−∆)kmu(x)/(1−|x|)m−1)=0, for0≤k≤p−1, whereD=BorB\{0}andhis a Borel measurable function onD×(0,∞)satisfying some appropriate conditions related to𝒥m,n(p).