A new Morse index theory for strongly indefinite functionals
Let \(H\) be a Hilbert space, \(A:H\to H\) a bounded linear selfadjoint operator with \(0\in\sigma(A)\) isolated. Let \(H_0=\text{Kern}(A)\), \(P^0:H\to H_0\) the orthogonal projection, \(P=I-P^0:H\to H^\perp_0\). It is allowed that \(H_0\) as well as the positive and negative eigenspaces of \(A\) are infinite-dimensional. Let \(G:H\to\mathbb R\) be of class \(C^1\) with \(PG':H\to H^\perp_0\) compact and \(G':H\to H\) quasi monotone in the following sense: If \(u_n\rightharpoonup u\), \(P u_n\to P_u\) then \(\limsup_{n\to\infty}\langle G(u_n)\), \(P^0(u_n-u)\rangle\leq 0\). The authors develop a Morse theory for the functional \(f(x)={{1}\over{2}}\langle Ax,x\rangle+G(x)\). In particular, they define the critical groups of an isolated critical point of \(f\) using finite-dimensional approximations. They also obtain Morse inequalities if \(f\) has only finitely many critical points. A related Morse theory for strongly indefinite functionals is due to \textit{W. Kryszewski} and \textit{A. Szulkin} [Trans. Am. Math. Soc. 349, No. 8, 3181--3234 (1997; Zbl 0892.58015)].
scientific article; zbMATH DE number 2102518