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 Δu+K(r)f(u)=0Δu+K(r)f(u)=0 on the exterior of the ball of radius R   centered at the origin in RNRN such that limr→∞⁡u(r)=0 with prescribed number of zeros where f:R→Rf:R→R is odd and there exists a β>0β>0 with f<0f<0 on (0,β)(0,β), f>0f>0 on (β,∞)(β,∞) with f superlinear for large u  , and K(r)∼r−αK(r)∼r−α with 0<α