Existence of Multiple Solutions for a Nonlinearly Perturbed Elliptic Parabolic System in R

We consider the following nonlinearly perturbed version of the elliptic-parabolic system of Keller-Segel type: ∂tu−∆u +∇ · (u∇v) = 0, t > 0, x ∈ R, −∆v + v − v = u, t > 0, x ∈ R, u(0, x) = u0(x) ≥ 0, x ∈ R, where 1 < p < ∞. It has already been shown that the system admits a positive solution for a small nonnegative initial data in L1(R2) ∩ L2(R2) which corresponds to the local minimum of the associated energy functional to the elliptic part of the system. In this paper, we show that for a radially symmetric nonnegative initial data, there exists another positive solution which corresponds to the critical point of mountain-pass type. The v-component of the solution bifurcates from the unique positive radially symmetric solution of −∆w + w = wp in R2.