In  the variational formula is obtained for reaction terms f(u)>0f(u)>0 in (0,1)(0,1) and the connection with the value c?c? introduced in  and  is established.
Lucia, Muratov and Novoga , in a more general frame of reaction–diffusion equation in infinite cylinders, obtain a variational characterization of c?c? for reaction terms as in  and  when (c?)2>4f′(0)(c?)2>4f′(0). Note that in these cases cM=c?cM=c?. Also they ME0328 prove the existence of a heteroclinic solution by minimizing a functional which involves the knowledge of c?c?. Their hypothesis (H3) implies (cM)2>4f′(0)(cM)2>4f′(0), been cMcM the supremum in (6). The existence of solution of the problem that they call (P′)(P′) enables to prove that in our case the value cMcM in (6) is attained.
In our framework, the study of the phase space allows to show that the profile that is obtained has a finite limit in −∞. This fact can only be established in  when the zeros of the reaction term f are isolated, see Corollary 6.8 in .