Existence of homoclinic solutions for nonlinear second-order coupled systems
In this work, the authors consider the second order coupled system on the real line \[ \begin{aligned} u''(t)- k_1^2 u(t)=f\big(t, u(t), v(t),u'(t), v'(t)\big), \;t\in \mathbb R, \\ v''(t)- k_2^2 v(t)=h\big(t, u(t), v(t),u'(t), v'(t)\big), \;t\in \mathbb R, \end{aligned}\eqno(1) \] where \( f, h : \mathbb R^5 \longrightarrow \mathbb R \) are \(L^1\)-Carathéodory functions, \( k_1, k_2 > 0\), the boundary conditions \[ \begin{aligned} u(\pm \infty) = u'(\pm \infty) = 0, \\ v(\pm \infty) = v'(\pm \infty) = 0. \end{aligned}\eqno(2) \] Under conditions on the monotonicity of the non-linearities \(f\) and \(h\) mainly, the authors establish the existence and the localization of a homoclinic solution for (1), that is, a non-trivial solution of the problem (1)--(2). The proof of the main result is topological based on the Schauder fixed-point theorem and a criterion of compactness, previously established by the first author. Also, it combines Green's functions technique with the notion of lower and upper solutions. The last section of the paper presents an application to a family of a coupled stationary nonlinear Schrödinger system (NLS).
scientific article; zbMATH DE number 6987665