Existence of multiple solutions for a nonlinearly perturbed elliptic parabolic system in $R^2$

We consider the following nonlinearly perturbed version of the elliptic-parabolic system of Keller-Segel type: $$displaylines{ partial_tu - Delta u+ abla cdot(u abla v)=0,quad t>0,; xinmathbb{R}^2, cr -Delta v+v-v^p=u,quad t>0,; xinmathbb{R}^2,cr u(0,x) =u_0(x)ge 0,quad xinmathbb{R}^2, }$$ where $1