Singular radial solutions for the Keller-Segel equation in high dimension

TL;DR

This paper investigates singular radially symmetric solutions of the super-critical Keller-Segel equation in high dimensions, revealing their role in bifurcation structures and the existence of oscillating regular solutions.

Contribution

It demonstrates the existence of singular solutions with multiple oscillations and their connection to regular solutions near bifurcation points in dimensions 3 to 9.

Findings

Singular solutions oscillate at least k times around equilibrium.

Regular solutions close to singular ones exist in dimensions 3 to 9.

Bifurcation branches oscillate and approach singular solutions in certain dimensions.

Abstract

We study singular radially symmetric solution of the stationary Keller-Segel equation, that is, an elliptic equation with exponential nonlinearity, which is super-critical in dimension $N \geq 3$. 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 $k \geq 0$, 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 $3 \leq N \leq 9$ there are regular solutions satisfying Neumann boundary conditions that are close to singular ones. Hence, it follows that there exist regular solutions on any ball with arbitrarily fast oscillations. For generic radii, we show that the bifurcation branches of regular solutions oscillate in the bifurcation plane when $3\leq N\leq 9$ and approach to a singular solution. In dimension $N > 10$, 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.