Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems
Consider the second order Hamiltonian system \[ \ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0\tag{HS}, \] where \(L\in C(R,R^N)\) is a symmetric matrix valued function and \(W \in C^1(R\times R^N, R)\). A nonzero solution \(u\) of (HS) is said to be homoclinic (to 0) if \(u(t)\rightarrow 0\) as \(|t|\rightarrow\infty\). The authors prove that problem (1) has infinitely many homoclinic orbits under the following conditions: {\parindent=8mm \begin{itemize}\item[(H] \(L \in C(R,R^{N{^2}})\) is a symmetric and positively definite matrix for all \(t\in R\) and there exists a continuous function \(l:R\rightarrow R\) such that \(l(t)>0\) for all \(t\in R\) and \[ (L(t)x,x)\geq l(t)|x|^2,\;l(t)\rightarrow\infty\;\text{as}\;|t|\rightarrow\infty. \] \item[(H] \(W(t,x)=a(t)|x|^r\), where \(a:R\rightarrow R^+\) is a continuous function such that \[ a\in L^{\frac{2}{2-r}}(R,R) \] and \(1<r<2\) is a constant. \end{itemize}} In fact, in Theorem 1.2 the condition respect to \(a\) is not sufficient and the condition that \(a\) is positive is also used in the proof of the Lemma 3.1. Theorem 1.2 in this paper generalizes the result in [\textit{Z. Zhang} and \textit{R. Yuan}, Nonlinear Anal., Theory Methods Appl. 71, No. 9, A, 4125--4130 (2009; Zbl 1173.34330)], in which \(a\) is a positive continuous function such that \[ a\in L^2(R, R)\cap L^{\frac{2}{2-r}}(R, R). \]
scientific article; zbMATH DE number 5804436