A positive equilibrium at the critical Dirichlet eigenvalue

We construct a connected unbounded uniformly smooth domain in $\R^{21}$ whose bottom Dirichlet eigenvalue is $1$, but on which $-\Delta u=u-u^2$ has a bounded positive solution with zero boundary values. Thus equality between the spectral threshold and the derivative of a strict KPP reaction at zero does not force extinction on all uniformly smooth domains. The domain is a slight widening of a slab of width $\pi$. Horizontal Hardy energy controls the transverse spectral deficit, while an explicit positive subsolution survives the quadratic reaction loss.