Existence of positive solution for a semi positone radial p-laplacian system.

Abstract

In this paper, we prove, for λλ and μμ large, the existence of a positive solution for the semi-positone elliptic system (P)⎧⎩⎨⎪⎪−Δpu=λω(x)f(v)−Δqv=μρ(x)g(u)(u,v)=(0,0)in Ω,in Ω,on ∂Ω,(P){−Δpu=λω(x)f(v)in Ω,−Δqv=μρ(x)g(u)in Ω,(u,v)=(0,0)on ∂Ω, where Ω=B1(0)={x∈RN:|x|≤1}Ω=B1(0)={x∈RN:|x|≤1}, and, for m>1m>1, ΔmΔm denotes the mm-Laplacian operator p,q>1p,q>1. The weight functions ωω, ρ:Ω¯¯¯¯→Rρ:Ω¯→R are radial, continuous, nonnegative and not identically null, and the non-linearities f,g:[0,∞)→Rf,g:[0,∞)→R are continuous functions such that f(t)f(t), g(t)≥−σg(t)≥−σ. The result presented extends, for the radial case, some results in the literature [D. D. Hai and R. Shivaji]. In particular, we do not impose any monotonic condition on ff or gg. The result is obtained as an application of the Schauder fixed point theorem and the maximum principle.