Existence for semilinear equations on exterior domains

In this paper we study radial solutions of Delta u + K (r) f (u) = 0 on the exterior of the ball of radius R > 0 centered at the origin in RN where f is odd with f < 0 on (0, beta), f > 0 on (beta, infinity), and f superlinear. The function K (r) is assumed to be positive and K (r) -> 0 as r -> infinity. We prove existence of an infinite number of radial solutions with u -> 0 as r -> infinity when K (r) similar to r(-alpha) with N < alpha < 2 (N - 1)