# Van der Waals--Allen--Cahn--Hilliard equation with a volume constraint

## TL;DR

This paper studies the multiplicity of solutions for a nonlinear elliptic equation of Van der Waals--Allen--Cahn--Hilliard type with a volume constraint, linking solution count to the domain's topological properties.

## Contribution

It provides new multiplicity results for solutions of a constrained nonlinear elliptic PDE, connecting solution counts to topological and homological invariants of the domain.

## Key findings

- Estimates the number of solutions based on domain topology.
- Establishes solution multiplicity for a class of nonlinear elliptic equations.
- Analyzes equations with asymmetric double well potentials.

## Abstract

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 $\Omega$.