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

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.

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