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

TL;DR

This paper investigates solutions to a stationary semilinear fourth-order Cahn--Hilliard equation in R^N, demonstrating the existence of multiple solutions via variational methods and numerical analysis, revealing a rich solution structure.

Contribution

It introduces a novel combination of variational and numerical methods to identify multiple solution families for the equation, extending previous results.

Findings

Existence of at least two solutions via Mountain Pass Lemma.

Countably many solutions established through Lusternik--Schnirel'man theory.

Numerical evidence suggests a much wider set of solutions than theoretically proven.

Abstract

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 are exponentially decaying at infinity, are studied. Using the Mounting Pass Lemma allows us the determination of two different solutions. 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.