Spectral-Edge Continuation, Dimensional Crossover, and Endpoint Stability for Partially Confined Nonlinear Schrödinger Equations

We study the global geometry and endpoint dynamics of positive solutions of $-Δu+|y|^2u+λu=g(u)$, $(y,z)\in\mathbb R^d\times\mathbb R^m$, $m=N-d$. Despite the loss of coercivity at $λ=-d$, we construct a connected component of centered positive solutions with frequency projection $(-d,\infty)$, assu