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

We study positive radial solutions to -Delta u = lambda K(vertical bar x vertical bar) f(u); x is an element of Omega(e) where lambda > 0 is a parameter, Omega(e) = {x is an element of R-N vertical bar vertical bar x vertical bar > r(0), r(0) > 0, N > 2}, A is the Laplacian operator, K is an element of C ([r(0), infinity), (0, infinity)) satisfies K(r) <= 1/r(N+u); mu > 0 for r >> 1, and f is an element of C-1 ([0, infinity), R) is a class of non -decreasing functions satisfying lim(s ->infinity) f (s)/s = infinity (superlinear) and f(0) < 0 (semipositone). We consider solutions, u, such that u -> 0 as vertical bar x vertical bar -> infinity, and which also satisfy the nonlinear boundary condition partial derivative u/partial derivative eta + <(c)over tilde>(u)u = 0 when vertical bar x vertical bar = r(0), where partial derivative/partial derivative eta is the outward normal derivative, and (c) over tilde is an element of C ([0, infinity), (0, infinity)). We will establish the existence of a positive radial solution for small values of the parameter lambda. We also establish a similar result for the case when u satisfies the Dirichlet boundary condition (u = 0) for vertical bar x vertical bar = r(0). We establish our results via variational methods, namely using the Mountain Pass Lemma. (C) 2015 Elsevier Inc. All rights reserved.