A Variant of Clark's Theorem and its Applications for Nonsmooth Functionals Without the Palais-Smale Condition
SIAM Journal on Mathematical Analysis
Society for Industrial and Applied Mathematics Publications
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.
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.