Singular radial solutions for the Lin-Ni-Takagi equation (Q2204087)

This article is concerned with the study of positive singular radial solutions of \(-\Delta u+u=u^p\) in \(B_R\setminus\{0\}\) subject to Neumann boundary condition \(\frac{\partial v}{\partial \nu}=0\) on \(\partial B_R\). Here \(B_R\) denotes the open unti ball in \({\mathbb R}^N\), \(N\geq 3\), with radius \(R>0\). The authors show that for any \(k\geq 1\) there is a singular solution that oscillates at least \(k\) times around a constant equilibrium. Further, it is obtained that the Morse index of the singular solution is finite or infinite if the exponent is respectively larger or smaller than the Joseph-Lundgren exponent.