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Existence and multiplicity of stationary solutions for a Cahn-Hilliard-type equation in \(\mathbb{R}^N\). (English) Zbl 1342.35108

Summary: Solutions of the stationary semilinear Cahn-Hilliard-type equation \[ -\Delta^2 u - u -\Delta(|u|^{p-1}u)=0 \quad \text{in } \mathbb{R}^N, \quad \text{with } p>1, \] 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.

MSC:

35J61 Semilinear elliptic equations
35G20 Nonlinear higher-order PDEs
35K52 Initial-boundary value problems for higher-order parabolic systems
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