Document Type

Article

Journal/Book Title/Conference

SIAM Journal on Mathematical Analysis

Volume

49

Issue

1

Publisher

Society for Industrial and Applied Mathematics Publications

Publication Date

2-16-2017

First Page

446

Last Page

470

DOI

10.1137/15M1034635

Abstract

By introducing a new notion of the genus with respect to the weak topology in Banach spaces, we prove a variant of Clark's theorem for nonsmooth functionals without the Palais-Smale condition. In this new theorem, the Palais-Smale condition is replaced by a weaker assumption, and a sequence of critical points converging weakly to zero with nonpositive energy is obtained. As applications, we obtain infinitely many solutions for a quasi-linear elliptic equation which is very degenerate and lacks strict convexity, and we also prove the existence of infinitely many homoclinic orbits for a second-order Hamiltonian system for which the functional is not in C1 and does not satisfy the Palais-Smale condition. These solutions cannot be obtained via existing abstract theory.

Comments

http://epubs.siam.org/doi/10.1137/15M1034635

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