  We give multiplicity results for the solutions of a nonlinear elliptic
equation, with an asymmetric double well potential of Van der
Waals-Allen--Cahn--Hilliard type, satisfying a linear volume constraint, on a
bounded Lipschitz domain $\Omega\subset\mathds R^N$. The number of solutions is
estimated in terms of topological and homological invariants of the underlying
domain $\Omega$.
