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On the Existence of Three Nontrivial Solutions of a Resonance Elliptic Boundary Value Problem with a Discontinuous Nonlinearity

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Abstract

We study the homogeneous Dirichlet problem for a second-order elliptic equation with a nonlinearity discontinuous in the state variable in the resonance case. A class of resonance problems that does not overlap with the previously investigated class of strongly resonance problems is singled out. Using the variational method, we establish a theorem on the existence of at least three nontrivial solutions of the problem under study (the zero is its solution). In this case, at least two nontrivial solutions are semiregular; i.e., the values of such solutions fall on the discontinuities of the nonlinearity only on a set of measure zero. We give an example of a nonlinearity satisfying the assumptions of this theorem. A sufficient semiregularity condition is obtained for a nonlinearity with subcritical growth at infinity, a case which is of separate interest. Applications of the theorem to problems with a parameter are considered. The existence of nontrivial (including semiregular) solutions of the problem with a parameter for an elliptic equation with a discontinuous nonlinearity for all positive values of the parameter is established.

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Authors and Affiliations

  1. Chelyabinsk State University, Chelyabinsk, 454001, Russia

    V. N. Pavlenko

  2. St. Petersburg State University, St. Petersburg, 199034, Russia

    D. K. Potapov

Authors
  1. V. N. Pavlenko
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  2. D. K. Potapov
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Correspondence to V. N. Pavlenko or D. K. Potapov.

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Translated by V. Potapchouck

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Pavlenko, V.N., Potapov, D.K. On the Existence of Three Nontrivial Solutions of a Resonance Elliptic Boundary Value Problem with a Discontinuous Nonlinearity. Diff Equat 56, 831–841 (2020). https://doi.org/10.1134/S0012266120070034

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  • DOI: https://doi.org/10.1134/S0012266120070034

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