Layered solutions with unbounded mass for the Keller–Segel equation

Abstract

In this paper, we construct several families of radial solutions for the stationary Keller–Segel equation. The first one consists in solutions which concentrate simultaneously on the interior and on the exterior boundary of an annulus. The second family is made of solutions on the unit ball which concentrate on an interior sphere while the solutions of the third one concentrate simultaneously on an interior sphere and on the boundary of the unit ball. We also show how to construct other families of multi-layered radial solutions in the unit ball under a suitable non degeneracy assumption. Our construction is based on a fixed point argument. Our work generalizes (Pistoia and Vaira in Proc R Soc Edinb Sect A 145(1):203–222, 2015) and continues the analysis started in Bonheure et al. (Multiple positive solutions of the stationary Keller–Segel system. arXiv:1603.07374v1, 2016).