Nonsmooth critical point theory and quasilinear elliptic equations

These lectures are devoted to a generalized critical point theory for nonsmooth functionals and to existence of multiple solutions for quasilinear elliptic equations. If f is a continuous function defined on a metric space, we define the weak slope |df|(u), an extended notion of norm of the Frechet derivative. Generalized notions of critical point and Palais-Smale condition are accordingly introduced. The Deformation Theorem and the Noncritical Interval Theorem are proved in this setting. The case in which f is invariant under the action of a compact Lie group is also considered. Mountain pass theorems for continuous functionals are proved. Estimates of the number of critical points of f by means of the relative category are provided. A partial extension of these techniques to lower semicontinuous functionals is outlined. The second part is mainly concerned with functionals of the Calculus of Variations depending quadratically on the gradient of the function. Such functionals are naturally continuous, but not locally Lipschitz continuous on H 0 1 . When f is even and suitable qualitative conditions are satisfied, we prove the existence of infinitely many solutions for the associated Euler equation. The regularity of such solutions is also studied.

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