Countable families of solutions of a limit stationary semilinear fourth-order Cahn-Hilliard-type equation I. Mountain pass and Lusternik-Schnirel’man patterns in RN

Abstract

Solutions of the stationary semilinear Cahn-Hilliard-type equation −Δ2u−u−Δ(|u|p−1u)=0 in RN, with p > 1, which are exponentially decaying at infinity, are studied. Using the mounting pass lemma allows us to determinate the existence of a radially symmetric solution. On the other hand, the application of Lusternik-Schnirel’man (L-S) category theory shows the existence of, at least, a countable family of solutions. However, through numerical methods it is shown that the whole set of solutions, even in 1D, is much wider. This suggests that, actually, there exists, at least, a countable set of countable families of solutions, in which only the first one can be obtained by the L-S min-max approach.