Estimates on the Green’s Function and Existence of Positive Solutions of Nonlinear Singular Elliptic Equations in the Half Space

Abstract

We establish a new 3G-Theorem for the Green's function for the half space

On the existence of positive continuous solutions for some polyharmonic elliptic systems on the half

“…Then the function q ∈ K ∞ m,n if and only if λ < 2m < µ. We note that for m = 1, the corresponding elliptic class K ∞ R n + := K ∞ 1,n R n + has been studied by Bachar and Mâagli in [1] for n ≥ 3 and by Bachar et al in [2] for n = 2.…”

On a parabolic problem with nonlinear term in a half space and global behavior of solutions

“…A Borel measurable function q in R n+1 belongs to the class P ∞ (R n ) if for all c > 0, Here and always G c (x, t, y, s) := 1 (t − s) n/2 exp −c |x − y| 2 t − s , for t > s and x, y ∈ R n .…”

“…As it is mentioned in the introduction the class P ∞ (R n + ) contains the time independent Kato class K ∞ (R n + ) (see [2] and [3]). In this section, we will show this claim and we provide some further examples of functions in P ∞ (R n + ).…”

“…Note that for n 2, we have by simple calculation that for each x, y ∈ R n + , 2 3 |x − y| 2 + x n y n |x − y| 2 + y 2 n 2 |x − y| 2 + x n y n . On the other hand, it was shown in [2], that for n 3 there exists a constant c > 0, such that 1 c min 1, x n y n |x − y| 2 1 |x − y| n−2 G(x, y) c min 1,…”

Existence results for some polyharmonic problems in the half-space

“…This form of the 3G-Theorem has been exploited to introduce the Kato class K ∞ m,n (see Definition 1.1), which was used in the study of some nonlinear polyharmonic equations (see [5] When m = 1, this class has been studied with its properties by Bachar and Mâagli [2] for n 3 and by Bachar et al [3] for n = 2.…”