Morse Theory for Continuous Functionals

Using the notion of weak slope introduced by M. Degiovanni and M. Marzocchi (1994, Ann. Mat. Pura Appl.167, 73-100), and also independently by C. Katriel (1994, Ann. Inst. H. Poincaré Anal. Non Linéaire11, 189-209), we obtain some results in Morse theory for continuous functionals having isolated critical points. The main tools are deformation properties generalizing classical ones for smooth functionals and refining those of J.-N. Corvellec, M. Degiovanni, and M. Marzocchi (1993, Topol. Methods Nonlinear Anal.1, 151-171). We obtain, for example, generalized Morse relations through the notion of critical group, which has a natural link with the weak slope. These results are developed in view of applications to non-smooth variational problems.