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Multiple solutions for strongly resonant nonlinear elliptic problems with discontinuities. (English) Zbl 1130.35033

Summary: We examine a nonlinear strongly resonant elliptic problem driven by the \(p\)-Laplacian and with a discontinuous nonlinearity. We assume that the discontinuity points are countable and at them the nonlinearity has an upward jump discontinuity. We show that the problem has at least two nontrivial solutions without using a multivalued interpretation of the problem as it is often the case in the literature. Our approach is variational based on the nonsmooth critical point theory for locally Lipschitz functions.

MSC:

35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
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