Homological local linking (Q5926225)

Let \(X\) be a Banach space and let \(F:X\to \mathbb{R}\) be a \(C^1\)-function with \(F(0)=0\) and let us suppose that 0 is an isolated critical point of \(F\).NEWLINENEWLINENEWLINEThe following general definition of homological local linking is introduced by the author: If \(q,\beta\) are positive integers one says that \(F\) has a local \((q,\beta)\)-linking near the origin if there exist a neighborhood \(U\) of 0 and subsets \(A,S,B\) of \(U\) with \(A\cap S=\emptyset\), \(0\notin A\), \(A\subset B\) and:NEWLINENEWLINENEWLINE1) 0 is the only critical point of \(F\) in \(F_0\cap U\), where \(F_0\) is the sublevel set \(\{u\in X:F(u)\leq 0\}\);NEWLINENEWLINENEWLINE2) denoting by \(i_{1^*}: H_{q-1} (A)\to H_{q-1} (U\setminus S)\) and \(i_{2**}:H_{q-1}(A \to H_{q-1}(B)\) the homology group homomorphisms induced by inclusions, \(\text{rank} i_{1^*}-\text{rank} i_{2^*}\geq\beta\)NEWLINENEWLINENEWLINE3) \(F\leq 0\) on \(B\);NEWLINENEWLINENEWLINE4) \(F>0\) on \(S\setminus \{0\}\).NEWLINENEWLINENEWLINEBy using this notion an existence result for critical points is given in Theorem 3.2. Some applications to second-order Hamiltonian systems are also included.