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Uniqueness Of Nonnegative Solutions For Semipositone Problems On Exterior Domains, Alfonso Castro, Lakshmi Sankar, Ratnasingham Shivaji

All HMC Faculty Publications and Research

We consider the problem

−Δu = λK(|x|)f(u), x∈Ω

u=0 if |x|=r₀

u→0 as |x|→∞,

where λ is a positive parameter, Δu = div(∇u)is the Laplacian of u, Ω = {x ∈ Rⁿ; n > 2,|x| > r₀}, K ∈ C¹([r₀,∞),(0,∞)) is such that lim r→∞ K(r) = 0 and f ∈ C¹([0,∞),R) is a concave function which is sublinear at ∞ and f(0) < 0. We establish the uniqueness of nonnegative radial solutions when λ is large.