Existence of critical-mass ground states

Determine whether the mass-constrained critical nonlinear Schrödinger energy on the product of Euclidean space and a compact metric graph admits a ground state at the threshold mass \(\mu_{\mathrm{ex}^*}(\mathbb R^N\times G,2_*)\), including graphs with a terminal edge or a cycle covering.

Background

For the $L^2$-critical exponent $p=2_* = 2+4/(N+1)$, the paper establishes graph-dependent thresholds separating finite and unbounded-below energy regimes. The threshold $\mu_{\mathrm{ex}^*}(\mathbb R^N\times G,2_*)$ lies between the critical masses associated with $\mathbb R^{N+1}_+$ and $\mathbb R^{N+1}$, and it coincides with the Euclidean critical mass when the graph admits a cycle covering and with the half-space critical mass when the graph contains a terminal edge.

The results prove existence below the threshold and nonexistence or unboundedness above appropriate comparison thresholds, but do not settle attainment exactly at $\mu=\mu_{\mathrm{ex}^*}(\mathbb R^N\times G,2_*)$.