Existence and nonexistence of sign-changing solutions for singular semilinear equations on exterior domains. (April 2022) - Iaia, Joseph A.
Abstract: In this paper we study solutions of Δ u + K ( | x | ) f ( u ) = 0 in the exterior of the ball of radius R > 0 in R N where f grows sublinearly at infinity and is singular at 0 with f ( u ) ∼ − 1 | u | q − 1 u, 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 .