Existence and nonexistence of sign-changing solutions for singular semilinear equations on exterior domains

Abstract

In this paper we study solutions of Δu+K(|x|)f(u)=0 in the exterior of the ball of radius R>0 in RN where f grows sublinearly at infinity and is singular at 0 with f(u)∼−1|u|q−1u, 0<q<1, for small u. We assume K(|x|)∼|x|−α for large |x| and establish existence of positive and sign-changing solutions when α>2(N−1) and R>0 is small. We prove nonexistence when R>0 is large and we also prove nonexistence for 2<α<N+q(N−2) and all R>0.