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

Abstract

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 C 1 and does not satisfy the Palais-Smale condition. These solutions cannot be obtained via existing abstract theory.

Multiple solutions for p(x)${p(x)}$‐curl systems with nonlinear boundary condition

“…In this section, we review some basic definitions and properties concerning the function spaces with variable exponent, and the space 𝐖 𝑝(𝑥) (Ω) of the divergence free vector function of 𝐋 𝑝(𝑥) (Ω), (see [3,7,17,18]).…”

Variational methods for degenerate Kirchhoff equations

“…Otherwise, the sequence of critical points {u k } +∞ k=1 does not necessarily converge to zero, see [17] for such an example. We also refer readers to [9,17,18] for certain improved versions of the Clark theorem and their various applications.…”