A multiplicity theorem for locally Lipschitz periodic functionals
The authors consider the periodic multivalued problem of the forced pendulum \[ \begin{cases} x''(t) + f(t)\in \biggl[\underline g\bigl(x(t) \bigr), \overline g \bigl(x(t) \bigr)\biggr], \quad \text{a.e. } t\in(0,1) \\ x(0) = x(1),\;x'(0) = x'(1) \end{cases} \] where \(f\in L^p(0,1)\), \(g\in L^\infty (\mathbb{R})\) is periodic of period \(T\), the conditions \(\int^T_0 g(s)ds = \int^1_0 f(t)dt =0\) hold and \(\underline g(s) = \lim_{\varepsilon \to 0^+} \text{ess inf}\{g (\sigma): |\sigma - s|< \varepsilon\}\), \(\overline g(s) = \lim_{\varepsilon \to 0^+} \text{ess sup} \{g(\sigma): |\sigma - s |< \varepsilon\}\). They show that the given problem admits at least two geometrically distinct solutions. The proof is based on a result of Lyusternik-Schnirelman type for locally Lipschitz functionals. Results in this direction were already obtained by \textit{K. C. Chang} [J. Math. Anal. Appl. 80, 102-129 (1981; Zbl 0487.49027)] through an extension of the deformation lemma. In contrast, the authors use the Ekeland variational principle.
scientific article; zbMATH DE number 861787