A multiplicity result for nonlinear second order periodic equations with nonsmooth potential
This article deals with the following periodic problem: \[ (| x'(t)| ^{p-2}x'(t))' \in \partial j(t,x(t)) \;\text{ almost everywhere on } [0,b], \quad x(0) = x(b), \;\;\;x'(0) = x'(b) \] with \(2 \leq p < \infty\) and a potential function \(j: [0,b] \times {\mathbb R} \to {\mathbb R}\) measurable in the first variable and locally Lipschitz in the second one (\(\partial\) represent the Clarke subdifferential). This problem is equivalent to analysis of critical points for the functional \[ \phi(x) = \frac1p \, \| x'\| _p^p + \int_0^b j(t,x(t))\, dt \] defined on \(W^{1,p}([0,b])\) and satisfying the nonsmooth \(PS_c\)-condition for any \(c \neq \int_0^b j_\pm(t) \, dt\) (\(j_\pm(t) = \lim_{\zeta \to \pm \infty} j(t,\zeta)\)) and bounded below under some natural conditions. The main result is the existence at least three distinct solutions. Some examples are presented.
scientific article; zbMATH DE number 2031082