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Positive solutions for a class of superlinear semipositone systems on exterior domains

Research paper by Abraham Abebe, Maya Chhetri, Lakshmi Sankar, R Shivaji

Indexed on: 25 Sep '14Published on: 25 Sep '14Published in: Boundary Value Problems



Abstract

We study the existence of a positive radial solution to the nonlinear eigenvalue problem −Δu=λK1(|x|)f(v)Open image in new window in ΩeOpen image in new window, −Δv=λK2(|x|)g(u)Open image in new window in ΩeOpen image in new window, u(x)=v(x)=0Open image in new window if |x|=r0Open image in new window (>0), u(x)→0Open image in new window, v(x)→0Open image in new window as |x|→∞Open image in new window, where λ>0Open image in new window is a parameter, Δu=div(∇u)Open image in new window is the Laplace operator, Ωe={x∈Rn∣|x|>r0,n>2}Open image in new window, and Ki∈C1([r0,∞),(0,∞))Open image in new window; i=1,2Open image in new window are such that Ki(|x|)→0Open image in new window as |x|→∞Open image in new window. Here f,g:[0,∞)→ROpen image in new window are C1Open image in new window functions such that they are negative at the origin (semipositone) and superlinear at infinity. We establish the existence of a positive solution for λ small via degree theory and rescaling arguments. We also discuss a non-existence result for λ≫1Open image in new window for the single equations case.MSC: 34B16, 34B18.