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Existence of infinitely many stationary layered solutions in \(\mathbb R^2\) for a class of periodic Allen-Cahn equations. (English) Zbl 1125.35342

Variational methods are used to find solutions of the semilinear elliptic equation \[ -\Delta u + a(x,y) W'(u)=0, \quad (x,y)\in\mathbb{R}^2 \] where \(a\) is periodic in both variables and \(W\) is a double-well potential. The main result states that there are infinitely many different (up to periodic translations in \(x\) and \(y\)) solutions. This is proved in three steps. First, the existence of solutions which are periodic in \(y\) and which decay to constant states as \(x\to\pm\infty\) is established. If there are only finitely many of these periodic solutions (up to translations in \(x\)) then in a second step the existence of heteroclinic orbits is shown which connect different periodic profiles as \(y\to\pm\infty\). In a last step, it is shown that if the number of these heteroclinic orbits is finite (up to translations in \(y\)) then there exist infinitely many multibump solutions.

MSC:

35J60 Nonlinear elliptic equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
35B10 Periodic solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
35J20 Variational methods for second-order elliptic equations
47J30 Variational methods involving nonlinear operators
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