On the existence of heteroclinic connections
Let \( W : \mathbb R^m \longrightarrow \mathbb R\) be a smooth nonnegative function that vanishes on a finite set \(A\), with \(\sharp A\geq 2\) and \(a_-, a_+ \) two distinct points in \(A\). The paper is concerned with the existence of heteroclinic connections between \(a_-\) and \( a_+\) for the equation \[ \ddot{u}+ \nabla W(u)=0, \;\;x \in \mathbb R,\tag{1} \] that is, solutions of (1) satisfying \[ \lim_{x\longrightarrow \pm\infty} u(x)= a_\pm. \] Under the following condition: \[ \sqrt{W(u)}\geq \rho(|u|),\quad |u|\geq r_0,\eqno{(H)} \] for some \(r_0>0\) and a nonnegative function \(\rho : [r_0, +\infty) \longrightarrow \mathbb R\) such that \(\int_{r_0}^{+\infty} \rho(r)dr =+\infty\), the authors prove that equation (1) possesses a solution defined on an interval \((l_-, l_+)\) with \(-\infty\leq l_-<0< l_+\leq +\infty\) and satisfies \[ \lim_{x\longrightarrow l_\pm} u(x)= a_\pm. \] The proof is based on the Hamiltonian principle of least action and consists in minimizing the functional defined on a suitable set of maps by \[ J(u)=\int_{x_1}^{x_2} \Big(\frac{1}{2}|\dot{u}|^2 + W(u)\Big) dx. \] The authors give some easy criteria to guarantee \(l_\pm=\pm\infty\).
scientific article; zbMATH DE number 6911817
scientific article