Existence of multiple solutions for a nonlinearly perturbed elliptic parabolic system in ℝ²

We consider the following nonlinearly perturbed version of the elliptic-parabolic system of Keller-Segel type: ∂_tu - Δu + ∇ = (u∇v) = 0, t > 0, x ∈ ℝ², -Δv +v -v^p = u, t > 0, x ∈ ℝ², u(0, x) = u ₀(x:) ≥ 0, x ∈ ℝ², 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 υ-component of the solution bifurcates from the unique positive radially symmetric solution of -Δw + w = w^p in ℝ².