A Variant of Clark's Theorem and Its Applications for Nonsmooth Functionals without the Palais--Smale Condition
Shaowei Chen, Zhaoli Liu, Zhi-Qiang Wang · SIAM Journal on Mathematical Analysis · 2017
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 $C^1$ and does not satisfy the Palais--Smale condition. These solutions cannot be obtained via existing abstract theory.