Bonheure, Denis; Casteras, Jean-Baptiste; Foldes, Juraj
(2020)
We study singular radially symmetric solution of the stationary Keller-Segel equation, that is, an elliptic equation with exponential nonlinearity, which is supercritical in dimension N >= 3. The solutions are unbounded at the origin and we show that they describe the asymptotics of bifurcation branches of regular solutions. It is shown that for any ball and any k >= 0, there is a singular solution that satisfies Neumann boundary condition and oscillates at least k times around the constant equilibrium. Moreover, we prove that in dimension 3 10, we show that the Morse index of the singular solution is finite, and therefore the existence of regular solutions with fast oscillations is not expected. (C) 2019 Elsevier Masson SAS. All rights reserved.