Existence and nonexistence of radial solutions for semilinear equations with bounded nonlinearities on exterior domains

Abstract

In this article 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 \({\mathbb R}^N\) where f is odd with f<0 on \(0, \beta\), f>0 on \(\beta\, \delta\) \(f\equiv 0  for (u> \delta\)), and where the function K(r) is assumed to be positive and \(K(r)\to 0) as (r \to \infty\)). The primitive \(F(u) = \int_0^u f(t)\), dt has a "hilltop'' at \(u=\delta\). With mild assumptions on f we prove that if \(K(r)\sim r^{-\alpha}\) with \(2< \alpha< 2(N-1\)) then there are n solutions of \(Delta\ u +(K(r)f(u)= 0\) on the exterior of the ball of radius R such that \(u\ to\ 0\) as \(r \to \infty\) if R>0 is sufficiently small. We also show there are no solutions if R>0 is sufficiently large. For more information see https://ejde.math.txstate.edu/Volumes/2020/117/abstr.html