Existence of solutions for semilinear problems with prescribed number of zeros on exterior domains

In this paper we prove the existence of an infinite number of radial solutions of Delta u + K(r) f(u) = 0 on the exterior of the ball of radius R centered at the origin in R-N such that lim(r ->infinity) u(r) = 0 with prescribed number of zeros where f : R -> R is odd and there exists a beta > 0 with f < 0 on (0, beta), f > 0 on (beta, infinity) with f superlinear for large u, and K(r) similar to r(-alpha) with 0 < alpha < N. (C) 2016 Elsevier Inc. All rights reserved.