A Variant of Clark's Theorem and its Applications for Nonsmooth Functionals Without the Palais-Smale Condition

Authors

Shaowei Chen, National Huaqiao University
Zhaoli Liu, Capital Normal University
Zhi-Qiang Wang, Utah State UniversityFollow

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

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

Recommended Citation

Chen, S., Liu, Z., Wang, Z.-Q. A variant of Clark's theorem and its applications for nonsmooth functionals without the Palais-Smale condition (2017) SIAM Journal on Mathematical Analysis, 49 (1), pp. 446-470.