About the Morse theory for certain variational problems
A short time after its inception Morse theory was carried over to the calculus of variations for single integrals. Multiple integrals, however, presented obstacles overcome only decades later by Palais and Smale. They assume a sort of weakened compactness property called the Palais-Smale condition. In this paper we show that no such condition is necessary for the class of problems we consider. By some a-priori estimates recently obtained by the author [see Math. Z. 186, 179-199 (1984)] we are able to artificially endow the space of admissible functions with the necessary compactness properties. We do this for variational problems with uniformly elliptic Euler equations of roughly the same type as those considered by \textit{O. A. Ladyzhenskaya} and \textit{N. N. Ural'tseva} [cf. Linear and quasilinear elliptic equations (1968; Zbl 0164.130), pp, 318- 336].
scientific article; zbMATH DE number 3853578
scientific article