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2015, 2015(special): 10-18. doi: 10.3934/proc.2015.0010

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

Pablo Álvarez-Caudevilla 1,

1. 

Universidad Carlos III de Madrid, Av. Universidad 30, 28911-Leganés, Spain

Received  September 2014 Revised  January 2015 Published  November 2015

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.
Keywords: variational setting, countable family of critical points., Stationary Cahn--Hilliard equation.
Mathematics Subject Classification: Primary: 35G20, 35K5.
Citation: Pablo Álvarez-Caudevilla. Existence and multiplicity of stationary solutions for a Cahn--Hilliard-type equation in $\mathbb{R}^N$. Conference Publications, 2015, 2015 (special) : 10-18. doi: 10.3934/proc.2015.0010
References:
[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 $\mathbbR^N$, ().   Google Scholar

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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. Google Scholar

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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). Google Scholar

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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. Google Scholar

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

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show all references

References:
[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 $\mathbbR^N$, ().   Google Scholar

[2]

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. Google Scholar

[3]

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

[4]

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. Google Scholar

[5]

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). Google Scholar

[6]

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. Google Scholar

[7]

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

[8]

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

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