Some remarks on the classification of global solutions with asymptotically flat level sets

We simplify some technical steps from \cite{S1} in which a conjecture of De Giorgi was addressed. For completeness we make the paper self-contained and reprove the classification of certain global bounded solutions for semilinear equations of the type $$\mathrm{△}u=W^{\mathrm{\prime }}(u),$$ where $W$ is a double well potential.