Double critical mass conjecture for radial Keller–Segel steady states

Let $\Omega=B_R(0)\subset\mathbb{R}^2$ with $R>0$ and $k>0$. A steady state is a solution $(u,v)$ of the stationary problem considered in the paper, with prescribed mass $\int_\Omega u=m$. The quantities $m_\star(2,R,k)$ and $m^\star(2,R,k)$ denote the two critical masses governing the numerical bifurcation diagram. Double critical mass conjecture. For these parameters, (i) there is a unique steady state with

$$\int_\Omega u=m$$

for each $m\in[0,m_\star(2,R,k)]$; (ii) there are two steady states with $\int_\Omega u=m$ for each $m\in(m_\star(2,R,k),m^\star(2,R,k))$; and (iii) there is a unique steady state with $\int_\Omega u=m^\star(2,R,k)$. This conjectural picture is motivated by simulations and describes a double critical mass phenomenon for the no-flux-Dirichlet Keller–Segel system; the statement concerns the multiplicity of steady states in the radial setting and is not established by the preceding existence result alone.