Title: Singular radial solutions for super-critical Keller-Segel and Lin-Ni-Takagi equation
Speaker: Juraj Foldes (University of Virginia)
Speaker's URL: https://uva.theopenscholar.com/juraj-foldes/
Abstract: We will discuss singular radially symmetric solution of the stationary Keller-Segel and Lin-Ni-Takagi equation, that is, an elliptic equation with exponential or power nonlinearity, which is super-critical in dimension bigger than 2. The solutions are unbounded at the origin, and we show that they describe the asymptotics of bifurcation branches of regular solutions. In particular, we will prove that for any ball and any positive k, there is a singular solution that satisfies Neumann boundary condition and oscillates at least k times around the constant equilibrium. Moreover, we will show 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. This is a joint work with Denis Bonheure (Université libre de Bruxelles) and Jean-Baptiste Casteras (Universidade de Lisboa).
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