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

publication date

  • September 2016

volume

  • 171

electronic international standard serial number (EISSN)

  • 1687-2770

abstract

  • Solutions of the stationary semilinear Cahn-Hilliard-type equation−Delta2u−u−Delta(|u|p−1u)=0in RN,with p>1,−Delta2u−u−Delta(|u|p−1u)=0in 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.

keywords

  • Stationary Cahn-Hilliard equation
    Variational setting
    Non-unique oscillatory solutions
    Countable family of critical points