Marco Degiovanni

A critical point theory for nonsmooth functional

SummaryA new generalized notion of ∥df(u)∥ is introduced, which allows to prove several results of critical point theory for continuous functionals. An application to variational inequalities is shown.

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.

Nontrivial Solutions for p-Laplace Equations with Right-Hand Side Having p-Linear Growth at Infinity

ABSTRACT The existence of a nontrivial solution for quasi-linear elliptic equations involving the p-Laplace operator and a nonlinearity with p-linear growth at infinity is proved. Techniques of Morse theory are employed.

Subdifferential Calculus and Nonsmooth Critical Point Theory

A general critical point theory for continuous functions defined on metric spaces has been recently developed. In this paper a new subdifferential, related to that theory, is introduced. In particular, results on the subdifferential of a sum are proved. An example of application to PDEs is sketched. Detailed applications to PDEs are developed in separate papers.

Homotopical properties of a class of nonsmooth functions

SummaryA class of extended real valued functionals, already considered for evolution problems, is studied. The set where the functional is finite is proved to be an absolute neighborhood extensor. Applications to critical point theory, involving Ljusternik-Schnirelman category and cohomological index, are shown. The stability under Γ-convergence of the homotopical type of the sublevels of the functional is also treated.

PERIODIC SOLUTIONS OF DYNAMICAL SYSTEMS WITH NEWTONIAN TYPE POTENTIALS

Periodic solutions of some singular dynamical systems are sought, under assumptions which include the case of Newtonian potential generated by a mass concentrated in a point. A result concerning solutions of minimal period is also given.

Euler equations involving nonlinearities without growth conditions

A functional is considered, whose Euler equation involves a nonlinearity without growth conditions. It is shown that every minimum point of the functional is a solution of the Euler equation in a suitable weak sense.

Perturbations of nonsmooth symmetric nonlinear eigenvalue problems

Abstract We consider a symmetric semilinear boundary value problem having infinitely many solutions. We prove that, if we perturb this problem in a non-symmetric way, then the number of solutions goes to infinity as the perturbation tends to zero. The growth conditions on the nonlinearities do not ensure the smoothness of the associated functional.

Multiple solutions of semilinear elliptic equations with one-sided growth conditions

Multiplicity results for semilinear elliptic equations are obtained under one-sided growth conditions on the nonlinearity. Techniques of nonsmooth critical point theory are employed.

On the Euler-Lagrange Equation for Functionals of the Calculus of Variations without Upper Growth Conditions

For a class of functionals of the calculus of variations, we prove that each minimum of the functional satisfies the associated Euler-Lagrange equation. The integrand is assumed to be convex, but no upper growth condition is imposed.