Multiplicity of layered solutions for Allen-Cahn systems with symmetric double well potential. (English) Zbl 1301.35033
Summary: We study the existence of solutions \(u : \mathbb{R}^3 \to \mathbb{R}^2\) for the semilinear elliptic systems
\[
- {\Delta} u(x, y, z) + {\nabla} W(u(x, y, z)) = 0, \eqno{(0.1)}
\]
where \(W : \mathbb{R}^2 \to \mathbb{R}\) is a double well symmetric potential. We use variational methods to show, under generic non-degenerate properties of the set of one dimensional heteroclinic connections between the two minima \(\mathbf{a}_\pm\) of \(W\), that (0.1) has infinitely many geometrically distinct solutions \(u \in C^2(\mathbb{R}^3, \mathbb{R}^2)\) which satisfy \(u(x, y, z) \to \mathbf{a}_\pm\) as \(x \to \pm \infty\) uniformly with respect to \((y, z) \in \mathbb{R}^2\) and which exhibit dihedral symmetries with respect to the variables \( y\) and \( z\). We also characterize the asymptotic behavior of these solutions as \(|(y, z) | \to + \infty\).
MSC:
| 35J50 | Variational methods for elliptic systems |
| 35J61 | Semilinear elliptic equations |
| 35J60 | Nonlinear elliptic equations |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35J20 | Variational methods for second-order elliptic equations |
| 34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |
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