Homoclinic solutions for a class of systems of second order differential equations
The authors prove the existence of a nontrivial homoclinic solution of the system \(\ddot q(t)+A(t)q(t)= -W_q(q(t),t)\), \(t\in\mathbb{R}^N\), \(t\in\mathbb{R}\), using the selfadjoint operator \(L\) given by \[ (Lq,p)= \int_\mathbb{R}(\dot q(t)\dot p(t)-A(t)q(t)p(t))dt, \qquad q,p\in H^1(\mathbb{R},\mathbb{R}^N). \] Assuming that (i) \(W\in C^1(\mathbb{R}^{N+1},\mathbb{R})\), \(A(t)\) is a continuous symmetric matrix, \(A(t)\) and \(W(q,t)\) are \(2\pi\)-periodic in \(t\); (ii) There is \(\alpha>2\) such that \(W_q(q,t)q\geq \alpha W(q,t)\), for each \(t\) and \(q\); (iii) There are \(a_1,a_2>0\) such that \(a_1|q|^\alpha\leq W(q,t)\) and \(|W_q(q,t)|\leq a_2|q|^{\alpha-1}\), for each \(t\) and \(q\); (iv) \(0\not\in \sigma(L)\); the existence of a nontrivial solution \(q\in C^2(\mathbb{R},\mathbb{R}^N)\cap H^2(\mathbb{R},\mathbb{R}^N)\) which is homoclinic to 0 is proved.
scientific article; zbMATH DE number 906864