Sharp threshold for PES free energy in dimension four

Let $N=4$ and let $\Omega$ be $\mathbb{R}^{4}$, $\mathbb{T}^{4}$, or a bounded $C^{2}$ Neumann domain. Let $\mathcal{A}_M(\Omega)$ denote the admissible class of densities of mass $M$, and let $\mathcal{F}_{\mathrm{PES}}$ be the PES free energy. Write $M_{\!*}=64\pi^{2}\tau/\chi$. Sharp-threshold conjecture. The PES free energy satisfies the alternative: (i) if $0<M<M_{\!*}$, then

$$\inf_{u\in\mathcal{A}_M(\Omega)}\mathcal{F}_{\mathrm{PES}}[u]>-\infty;$$

(ii) if $M>M_{\!*}$, then

$$\inf_{u\in\mathcal{A}_M(\Omega)}\mathcal{F}_{\mathrm{PES}}[u]=-\infty;$$

and (iii) at $M=M_{\!*}$, loss of compactness occurs through one-point mass concentration, with the precise compactness alternative depending on the domain and boundary conditions. On $\mathbb{R}^{4}$, the usual tightness or moment condition is imposed to control the entropy from below. This conjecture is motivated by the predicted sharp logarithmic Hardy–Littlewood–Sobolev/Adams constant $a_{\sharp}(M,\tau,\Omega)=M/(32\pi^{2}\tau)$ and the resulting candidate critical mass; the boundedness and concentration assertions remain to be established in the stated generality.