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

Pablo Álvarez-Caudevilla

doi:10.3934/proc.2015.0010

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2015, Volume 2015: 10-18. Doi: 10.3934/proc.2015.0010 open access

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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

Received: August 31, 2014

Revised: December 31, 2014

Published: October 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 35G20, 35K52.

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References

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