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Nonsmooth critical point theory and quasilinear elliptic equations

  • Annamaria Canino
  • Marco Degiovanni
Chapter
Part of the NATO ASI Series book series (ASIC, volume 472)

Abstract

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 Fréchet 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.

Keywords

Weak Solution Variational Inequality Open Cover Critical Point Theory Quasilinear Elliptic Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • Annamaria Canino
    • 1
  • Marco Degiovanni
    • 2
  1. 1.Dipartimento di MatematicaUniversità della CalabriaArcavacata di RendeItaly
  2. 2.Dipartimento di MatematicaUniversità CattolicaBresciaItaly

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