A note on the resonance case for asymptotically linear wave equations (Q1059221)

The paper is mainly concerned with the multiple periodic solutions to the problem of the wave equation in the resonance case: \(u_{tt}- u_{xx}+g(x,t,\xi)=0\) on \((0,\pi)\times {\mathbb{R}}\), with \(u(0,t)=u(\pi,t)=0\) and \(u(x,t+2\pi)=u(x,t),\) where \(g(x,t,\xi)\) is strictly increasing, continuously differentiable and odd in \(\xi\), \(2\pi\)-periodic in t. Assume that \(\lim_{| t| \to \infty}g(x,t,\xi)/\xi =b\) and \(g=b\xi +g_ 1\); \(\lim_{| t| \to 0}g(x,t,\xi)/\xi =+\infty\), where b may be equal to an eigenvalue of the operator \(A=\partial^ 2/\partial t^ 2-\partial^ 2/\partial x^ 2.\) The authors improve and extend the results of Wu. They prove that without additional convex and concave conditions on g at \(\xi =0\) there still exist infinitely many solutions of the problem in the following cases: (i) \(b\in (0,\infty)\), \(g_ 1\) bounded; (ii) \(b=0\), and \(| g(x,t,\xi)| <\gamma \xi +c\), \(\gamma <3\); (iii) \(b\in (0,\infty)\) and \(c_ 1\xi^{\alpha}-c_ 2<g_ 1(x,t,\xi)<c_ 3\xi^{\alpha}+c_ 4,\) \(\forall \xi >0\), where \(c_ i>0\) and \(\alpha <1.\) The basic tools are the variational method in critical point theory and the index theories. An integral estimate and the truncation technique are used to obtain the improvement.