Limit Cases of the Palais-Smale Condition
Condition (P.-S.) may seem rather restrictive. Actually, as Hildebrandt [4; p. 324] records, for quite a while many mathematicians felt convinced that inspite of its success in dealing with one-dimensional variational problems like geodesics (see Birkhoff’s Theorem I.4.4, for example, or Palais’  work on closed geodesics), the Palais-Smale condition could never play a role in the solution of “interesting” variational problems in higher dimensions.
KeywordsRiemannian Surface Dirichlet Problem Nontrivial Solution Limit Case Homotopy Class
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