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Solutions of the stationary semilinear Cahn--Hilliard equation  -\Delta^2 u - u -\Delta(|u|^{p-1}u)=0 in R^N, with p>1, which areexponentially decaying at infinity, are studied. Using the Mounting Pass Lemmaallows us the determination of two different solutions. On the other hand, theapplication of Lusternik--Schnirel'man (L--S) Category Theory shows theexistence of, at least, a countable family of solutions.  However, through numerical methods it is shown that the whole set ofsolutions, even in 1D, is much wider. This suggests that, actually, thereexists, at least, a countable set of countable families of solutions, in whichonly the first one can be obtained by the L--S min-max approach.

Countable families of solutions of a limit stationary semilinear  fourth-order Cahn--Hilliard equation I. Mountain pass and  Lusternik--Schnirel'man patterns in R^N