Existence of positive radial solutions for superlinear, semipositone problems on the exterior of a ball

We study positive radial solutions to −Δu=λK(|x|)f(u); x∈Ωe where λ>0 is a parameter, Ωe={x∈RN‖x|>r0,r0>0,N>2}, Δ is the Laplacian operator, K∈C([r0,∞),(0,∞)) satisfies K(r)≤1rN+μ;μ>0 for r>>1, and f∈C1([0,∞),R) is a class of non-decreasing functions satisfying lims→∞⁡f(s)s=∞ (superlinear) and f(0)<0 (semipositone). We consider solutions, u, such that u→0 as |x|→∞, and which also satisfy the nonlinear boundary condition ∂u∂η+c˜(u)u=0 when |x|=r0, where ∂∂η is the outward normal derivative, and c˜∈C([0,∞),(0,∞)). We will establish the existence of a positive radial solution for small values of the parameter λ. We also establish a similar result for the case when u satisfies the Dirichlet boundary condition (u=0) for |x|=r0. We establish our results via variational methods, namely using the Mountain Pass Lemma.