Positive radial solutions of n-dimensional elliptic systems with indefinite weight functions and n parameters

Abstract

Under simple conditions on f and a, we show the existence of positive radial solutions for the n-dimensional elliptic differential system u(x) + Λa(|x|)f(u(x)) = 0, R 1 < |x| < R 2 , u| |x|=R 1 = u| |x|=R 2 = 0.

Existence, Nonexistence and Multiplicity Results for Positive Radial Solutions of n−dimensional Elliptic System

“…Referring to [1], we can obtain the existence of a positive solution of system (1,3) is equivalent to the fixed-point system for all s ∈ J, and so…”

“…This paper considers the existence, nonexistence and multiplicity of positive solutions to the following n−dimensional elliptic system { ∆u(x) + Λa(|x|)f(u(x)) = 0 in Ω, u = 0 on ∂Ω, (1.1) where Ω = {x ∈ R n : R1 < |x| < R2}, R1 > R2 > 0, n ≥ 2, a(|x|) is allowed to change sign on Refferring to [1], by a positive solution of (1.1) is meant a solution u ∈ C 2 (Ω) with u ≥ 0 in Ω. By the strong maximum principle, it follows that u > 0 in Ω.…”

“…However, there are relatively few studies on n−dimensional elliptic systems. In [1], Feng and Li studied the existence and multiplicity of positive solutions for (1.1) by Fixed point theorem of cone expansion and compression of norm type. In [25], Fotouhi, Shahgholian and Weiss by the epiperimetric inequality approach studied the following system…”