On periodic solutions of a class of second order nonautonomous systems with nonhomogeneous potentials indefinite in sign
The authors consider the system \[ \ddot x(t)+ b(t)V'(x(t))= 0, \] where \(V= V_1+\cdots+ V_m\), \(V_i\in C^2(\mathbb{R}^N)\) \((N\geq 2)\), \(i=1,\dots, m\), \(b\in C(\mathbb{R},\mathbb{R}^N)\), \(b(t+ T)= b(t),\) \(\forall t\in\mathbb{R}\) \((T>0)\) and they prove the following. Let \(V_i\), \(i=1,\dots, m\) be positively homogeneous with the degrees \(\beta_1<\cdots< \beta_m\), \(\beta_1> 2\), \(\beta_{m-1}< 6\), \(\beta_m\geq 8\) and \(V_i(x)> 0\), \(i= 1,\dots,m\), \(x\neq 0\), \(V_m\) satisfying: \(V'_m(x)\) is positive definite for \(x\neq 0\). If \(b\) satisfies \((\text{b}_1)\) \(\int^T_0 b(t)<0\); \((\text{b}_2)\) \(b^+(t)\in H^1([0, T])\), \((b^+)\in L^{3/2}([0,T])\); \((\text{b}_3)\) \(Z^+\neq \emptyset\) is finite, where \(Z^+= \{t\in[0, T]: b(t)= 0\) and \(\exists\varepsilon= \varepsilon(t)>0\) so that \(b(s_1)b(s_2)> 0\) for \(s_1\in (t-\varepsilon, t)\), \(s_2\in(t,t+ \varepsilon)\}\); then there exists at least one nonzero \(T\)-periodic solution of the given system. A similar result is proved for \[ \ddot x(t)- A(t)x(t)+ b(t)V'(x(t))= 0, \] where \(A= (A_{ij})\), \(A_{ij}\in C(\mathbb{R})\), \(A_{ij}(t+ T)= A_{ij}(t)\), \(\forall t\in\mathbb{R}\), \(i,j= 1,\dots, N\) is a positive definite symmetric matrix for any \(t\in\mathbb{R}\), in \((\text{b}_1)<\) being replaced by \(\neq\). The proofs are based on a finite-dimensional approximation, a truncation argument of \(V\) and the use of Morse index estimates for critical points of mountain pass type.
scientific article; zbMATH DE number 1089778