Estimates on the Green Function and Existence of Positive Solutions of Nonlinear Singular Elliptic Equations

Abstract

We establish a 3G-Theorem for the Green's function for an unbounded regular domain D in R n (n ≥ 3), with compact boundary. We exploit this result to introduce a new class of potentials K(D) that properly contains the classical Kato class K ∞ n (D). Next, we study the existence and the uniqueness of a positive continuous solution u inD of the following nonlinear singular elliptic problem      ∆u + ϕ(., u) = 0 , in D(in the sense of distributions)where ϕ is a nonnegative Borel measurable function in D × (0, ∞), that belongs to a convex cone which contains, in particular, all functions ϕ(x, t) = q(x)t −σ , σ ≥ 0 with q ∈ K(D). We give also some estimates on the solution u.

Existence of positive bounded solutions for some nonlinear elliptic systems

“…Since h(x) = 1 − 1 |x| n−2 ∼ |x|−1 |x| , it follows from the choice of σ , γ and Proposition 3.4 in [2] that…”

“…where λ, μ are nonnegative constants, the functions f, g : (0, ∞) → [0, ∞) are continuous and the functions p, q are nonnegative in B(D) satisfying some hypotheses related to the Kato class introduced and studied in [2] and [7]. More precisely, we will give two existence results for (1.2) as f and g are nondecreasing or nonincreasing.…”

Positive solutions for some competitive elliptic systems

“…Ò Ø ÓÒ 1.3º (see [5] and [22] We remark that in the case where D is bounded and if d denotes its diameter,…”

“…The following compactness result will be used and it is proved in [22] for bounded domains and in [5] for unbounded ones.…”

“…where α ≥ 1, β ≥ 1, r ≥ 0, s ≥ 0 and p, q are two nonnegative functions in the Kato class K(D) introduced and studied in [5] and [22].…”