Topological degree of critical points of 2-dimension mountain-pass type produced by 2-dimension link (Q1178395)

Let \(H\) be a real Hilbert space, \(U\subset H\) an open subset and \(f\in C^ 2(U,\mathbb{R})\) with \(f'=I-T\) where \(T\) is a compact map. Let \(u_ 0\in U\) be a 2-dimensional Mountain-Pass Type critical point of \(f\). The author communicates to have shown under suitable conditions on some eigenvalues of \(D^ 2f(u_ 0)\), that \(\hbox{ind}(f',u_ 0)=1\), where \(\hbox{ind}(f',u_ 0)\) stands for the Leray-Schauder index of \(f'\) at \(u_ 0\).