Brake orbits type solutions to some class of semilinear elliptic equations (Q2641913)

In this paper the authors continue their study of planar equations of the form \[ - \Delta u (x,y) + a(x) W'(u(x,y)) = 0\,, \qquad (x,y) \in \mathbb R^2, \tag \(*\) \] under the following assumptions: (i) \(a: \mathbb R \rightarrow \mathbb R\) is Hölder continuous, \(T\)-periodic, and satisfies \(0 < \min_{x \in \mathbb R} \min_{a(x)}a(x)< \max_{x \in \mathbb R}\max_{a(x)} a(x) \), (ii) \(W \in C^2(\mathbb R)\) satisfies \(W(s) \geq 0\), \(\forall s \in \mathbb R\), \(W(s) > 0\), \(\forall s \in ]-1, 1[\), \(W(\pm 1) = 0\) and \(W''(\pm 1)>0\). For instance, \(W\) could be a Ginzburg-Landau or a sine-Gordon potential. The solutions are also subject to the side conditions \[ | u(x,y)| \leq 1, \qquad (x,y) \in \mathbb R^2, \] and \[ \lim_{x \to \pm \infty} u(x,y) = \pm 1, \] uniformly with respect to \(y \in \mathbb R\). A variational method carefully ``taylored'' to the problem. It is used to construct infinitly many geometrically distinct classical solutions. Here, a key role is played by the associated one-dimensional problem \[ \begin{aligned}- q''(x) + a(x) W'(q(x)) 0, &\qquad x \in \mathbb R,\\ | q(x)| \leq 1, &\qquad x \in \mathbb R,\\ \lim_{x \to \pm\infty} q(x) = \pm 1.&{} \end{aligned} \] plays a key role. The solutions of this problem are the stationary points of the action functional \[ F(q) := \int_\mathbb R \frac12 | \dot{q}(x)| ^2 + a(x) W(q(x)): d x \] on the space \(\Gamma := z_0 + H^1(\mathbb R)\), where \(z_0 \in C^\infty(\mathbb R)\) is a fixed increasing function satisfying \(z_0(x) = \pm 1\) for \(\pm x \geq 1\). A discreteness condition concerning the set of minimizers of \(F\) on \(\Gamma\) is required, and various sufficient conditions on the coefficient \(a(x)\) are indicated, under which this discreteness condition is known to be satisfied. Solutions of the original problem are viewed as trajectories of the infinite-dimensional dynamical system generated by \[ \frac{d^2}{d y^2} u(\cdot, y) = F'(u(\cdot,y)) \] in the configuration space \(\Gamma\), with \(y\) playing the role of time. The ``energy'' \[ E_{u}(y) := - \tfrac12 \| \partial_y u(\cdot, y)\| ^2_{L^2(\mathbb R)} + F(u(\cdot, y)) \] is therefore conserved along such a trajectory. It is shown that for every \(c_p\) belonging to a dense open subset of some non-empty open interval \(]c, c^*[\) there exists a ``brake type orbit'' of energy \(c_p\), i.e. a periodic trajectory, oscillating between two distinct points of \(\Gamma\). The solutions of \((*)\) corresponding to these trajectories are therefore periodic in the \(y\)-variable (the period depending on the energy \(c_p\)), and for each \(c_p\) the plane is divided into horizontal strips of equal width, on which the corresponding solution satisfies Neumann boundary conditions.