Stability of equilibria of an aggregation-diffusion energy on sphere

Abstract

We consider an aggregation-diffusion energy on the sphere and investigate the stability of its equilibria. The energy consists of a porous-medium type nonlinear entropy $\frac{1}{m-1}\int \rho (x{)}^m\mathrm{d}S(x)$ with $m>1$, together with an interaction energy modeled by a quadratic interaction potential. The energy generalizes the Onsager free energy with dipolar potential, which models polymer orientation. Our study complements the authors' previous work [Nonlinearity {\bf 39} (2026), 055011], where the ground states of the energy functional were investigated. In the current paper we extend the previous results on existence of equilibria for $m>2$ from $d=2$ to arbitrary dimension $d\ge 2$. This allows us to present the bifurcation structure (with respect to the interaction strength) of all equilibria in any dimension $d\ge 2$, for all $m>1$. Furthermore, we provide a complete classification of the stability of all equilibria of the energy, by deriving a criterion for stability that can be checked explicitly. In particular, for $m>2$ we identify a saddle-node bifurcation of strictly supported equilibria, and a subcritical pitchfork bifurcation where the uniform distribution loses stability.