Abstract
We establish a new form of the3Gtheorem for polyharmonic Green function on the unit ball ofℝn(n≥2)corresponding to zero Dirichlet boundary conditions. This enables us to introduce a new class of functionsKm,ncontaining properly the classical Kato classKn. We exploit properties of functions belonging toKm,nto prove an infinite existence result of singular positive solutions for nonlinear elliptic equation of order2m.
Highlights
In [2], Boggio gave an explicit expression for the Green function Gm,n of (− )m on the unit ball B of Rn (n ≥ 2) with Dirichlet boundary conditions u
For the case n ≥ 3, Zhao proved in [19] the existence of a positive constant Cn such that for each x, y, z in Ω, G1,n(x, z)G1,n(z, G1,n(x, y)
A Borel measurable function φ in Ω belongs to the Kato class Kn(Ω) if φ satisfies the following conditions: lim α→0 sup x∈Ω
Summary
716 Singular solutions for polyharmonic equation obtained the following inequality called 3G theorem: there exists a constant am,n > 0 such that for each x, y, z ∈ B, Gm,n(x, z)Gm,n(z, Gm,n(x, y) The Green function for the Laplacian (m = 1) satisfies the above inequality in an arbitrary bounded C1,1 domain Ω in Rn. for the case n ≥ 3, Zhao proved in [19] the existence of a positive constant Cn such that for each x, y, z in Ω, G1,n(x, z)G1,n(z, G1,n(x, y)
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