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Existence and multiplicity of stationary solutions for a Cahn--Hilliard-type equation in $\mathbb{R}^N$

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  • Solutions of the stationary semilinear Cahn--Hilliard-type equation $$ -\Delta^2 u - u -\Delta(|u|^{p-1}u)=0 \quad \mbox{in}     \mathbb{R}^N, \quad \mbox{with} \quad p>1, $$ which are exponentially decaying at infinity, are studied. Using the Mounting Pass Theorem 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.
    Mathematics Subject Classification: Primary: 35G20, 35K52.


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  • [1]

    P. Álvarez-Caudevilla, J.D. Evans and V.A. Galaktionov Countable families of solutions of a limit stationary semilinear fourth-order Cahn-Hilliard-type equation I. Mountain Pass Theorem and Lusternik-Schnirel'man patterns in $\mathbbR^N$, submitted.


    P. Álvarez-Caudevilla and V.A. Galaktionov, Steady states, global existence and blow-up for fourth-order semilinear parabolic equations of Cahn-Hilliard type, Advances Nonl. Stud., 12 (2012), 315-361.


    A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.


    J.D. Evans, V.A. Galaktionov and J.F. Williams, Blow-up and global asymptotics of the limit unstable Cahn-Hilliard equation, SIAM J. Math. Anal., 38 (2006), 64-102.


    V.A. Galaktionov, E. Mitidieri and S.I. Pohozaev, Variational approach to complicated similarity solutions of higher-order nonlinear PDEs. II, Nonl. Anal.: RWA, 12 (2011), 2435-2466 (arXiv:1103.2643).


    G.A. Klassen and E. Mitidieri, Standing wave solutions for a system derived from the FitzHugh-Nagumo equations for nerve conduction, SIAM J. Math. Anal., 17 (1986), 74-83.


    A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives, Adv. Math. Sci. Appl., 8 (1998), 965-985.


    L.A. Peletier and W.C. Troy, Spatial Patterns. Higher Order Models in Physics and Mechanics, Birkhäusser, Boston/Berlin, 2001.

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