Some rigidity results and asymptotic porperties for solutions to semilinear elliptic P.D.E

We will present some rigidity results for solutions to semilinear elliptic equations of the form $\text{Deltau}=W^{\mathrm{\prime }}(u)$, where W is a quite general potential with a local minimum and a local maximum. We are particularly interested in Liouvlle-type theorems and symmetry results, which generalise some known facts about the Cahn-Hilliard equation.