EXISTENCE AND NONEXISTENCE OF SOLUTIONS FOR SUBLINEAR PROBLEMS WITH PRESCRIBED NUMBER OF ZEROS ON EXTERIOR DOMAINS

We prove existence 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 if R > 0 is sufficiently small. We assume f : R -> R is odd and there exists a beta > 0 with f < 0 on (0, beta), f > 0 on (beta, infinity) with f sublinear for large u, and K(r) similar to r(-alpha) for large r with alpha > 2(N - 1). We also prove nonexistence if R > 0 is sufficiently large.