Ljusternik-Schnirelman theory with local Palais-Smale condition and singular dynamical systems (Q1182847)

The author proves the existence of infinitely many \(T\)-periodic solutions for second order systems \[ \ddot u+a\cdot u+W'(t,u)=0 \] with singular potential \(W\simeq -1/| x|^ \alpha\), \(\alpha\geq 2\), at \(x=0\) where in contrast to the known results (\(a=0\), strong growth conditions on \(W(t,x)\) and \(W'(t,x):=\nabla_ xW(t,x)\) as \(| x|\to \infty\)) the coefficient \(a\) now has to satisfy \(a<(\pi/T)^ 2\) and the potential \(W(t,x)\) as well as the function \(W'(t,x)\cdot x-2W(t,x)\) are allowed to grow of order \(| x|^ \Theta\), \(\Theta < 2\), for large values of \(x\). This more general situation is handled with the help of an appropriate Ljusternik-Schnirelman type theorem.