= Solution
At fixed particle number $N_p=\int|\psi|^2d\mathbf x$, an equilibrium <wavefunction> is a constrained critical point of the <energy functional>. Introduce the <chemical potential> $\mu$ as the <Lagrange multiplier> for $N_p$, and impose $\delta(E-\mu N_p)=0$. The resulting stationary <cubic-quintic Gross–Pitaevskii equation> is
$$
\boxed{-\frac{\hbar^2}{2m}\nabla^2\psi_0
+\left(V_0|\psi_0|^2+W_0|\psi_0|^4\right)\psi_0=\mu\psi_0}.
$$
The time-dependent equilibrium has $\psi(\mathbf x,t)=e^{-i\mu t/\hbar}\psi_0(\mathbf x)$, so its <number density> is stationary even though its <quantum phase> changes uniformly. The specified particle number fixes $\mu$ together with the boundary conditions. In an infinite uniform bulk, particle number is understood through a finite volume or a prescribed bulk density.
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