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$p$-Laplacian; nonsmooth critical point theory; Clarke subdifferential; saddle point theorem; periodic solution; Poincare-Wirtinger inequality; Sobolev inequality; nonsmooth Palais-Smale condition
In this paper we examine nonlinear periodic systems driven by the vectorial $p$-Laplacian and with a nondifferentiable, locally Lipschitz nonlinearity. Our approach is based on the nonsmooth critical point theory and uses the subdifferential theory for locally Lipschitz functions. We prove existence and multiplicity results for the “sublinear” problem. For the semilinear problem (i.e. $p = 2$) using a nonsmooth multidimensional version of the Ambrosetti-Rabinowitz condition, we prove an existence theorem for the “superlinear” problem. Our work generalizes some recent results of Tang (PAMS 126(1998)).
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