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

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 ≥ 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 ≤ N ≤ 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 ≤ N ≤ 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.

Singular radial solutions for the Lin–Ni–Takagi equation

“…We remark that an analogous result with u p replaced by λe u (with λ as a bifurcation parameter) been obtained by the authors and Bonheure in [3].…”

“…Also, without additional information one cannot combine Theorem 1.5 and Theorem 1.3 to prove Theorem 1.4 by limiting procedure. We remark that the oscillations and convergence of B i was proved by authors and Bonheure in [3] for (1.1) with v p replaced by λe v . The proof in the present case is more involved and will be published separately.…”

“…It is crucial to control the size of the neighbourhood with respect to parameter p. The proof is rather technical and requires very detailed information about solutions. Unlike in [3], our estimates cease to hold before the first intersection point with 1 that we denote r p . At least heuristically r p ≈ 1 √ p (in fact the upper bound can be made rigorous).…”

“…Finally, we briefly sketch main ideas of the proofs. To prove Theorem 1.4, we follow the general framework used in [3]. Specifically, Theorem 1.4 is a consequence of continuity of the function p → R i p for all i ∈ N and…”

“…To establish of (1.8), as in [3], we obtain very precise estimates of U * p in a neighbourhood of the origin. It is crucial to control the size of the neighbourhood with respect to parameter p. The proof is rather technical and requires very detailed information about solutions.…”