Multiple positive solutions of the stationary Keller-Segel system

作者：Bonheure Denis; Casteras Jean Baptiste; Noris Benedetta^*

来源：Calculus of Variations and Partial Differential Equations, 2017, 56(3): 74.

DOI：10.1007/s00526-017-1163-3

We consider the stationary Keller-Segel equation
{-Lambda v + v = lambda e(v), v > 0 in Omega,
partial derivative(nu)v = 0 on partial derivative Omega,
where Omega is a ball. In the regime lambda -> 0, we study the radial bifurcations and we construct radial solutions by a gluing variational method. For any given n is an element of N-0, we build a solution having multiple layers at r(1), ... , r(n) by which we mean that the solutions concentrate on the spheres of radii r(i) as lambda -> 0 (for all i = 1, ... , n). A remarkable fact is that, in opposition to previous known results, the layers of the solutions do not accumulate to the boundary of Omega as lambda -> 0. Instead they satisfy an optimal partition problem in the limit.

• 出版日期2017-6