Abstract:
This paper is concerned with the existence of semiregular solutions to the Dirichlet problem for an equation of elliptic type with discontinuous nonlinearity and when the differential operator is not assumed to be formally self-adjoint. Theorems on the existence of semiregular (positive and negative) solutions for the problem under consideration are given, and a principle of upper and lower solutions giving the existence of semiregular solutions is established. For positive values of the spectral parameter, elliptic spectral problems with discontinuous nonlinearities are shown to have nontrivial semiregular (positive and negative) solutions.
Bibliography: 32 titles.
Keywords:
spectral problems, equations of elliptic type, discontinuous nonlinearity, semiregular solutions, the method of upper and lower solutions.
Citation:
V. N. Pavlenko, D. K. Potapov, “The existence of semiregular solutions to elliptic spectral problems with discontinuous nonlinearities”, Sb. Math., 206:9 (2015), 1281–1298
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\paper The existence of semiregular solutions to elliptic spectral problems with discontinuous nonlinearities
\jour Sb. Math.
\yr 2015
\vol 206
\issue 9
\pages 1281--1298
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Linking options:
https://www.mathnet.ru/eng/sm8427
https://doi.org/10.1070/SM2015v206n09ABEH004496
https://www.mathnet.ru/eng/sm/v206/i9/p121
This publication is cited in the following 14 articles:
V. N. Pavlenko, D. K. Potapov, “Semi-regular solutions of integral equations with discontinuous nonlinearities”, Math. Notes, 116:1 (2024), 93–103
O. V. Baskov, D. K. Potapov, “Existence of Solutions to the Non-Self-Adjoint Sturm–Liouville Problem with Discontinuous Nonlinearity”, Comput. Math. and Math. Phys., 64:6 (2024), 1254
O. V. Baskov, D. K. Potapov, “Existence of solutions to the non-self-adjoint Sturm–Liouville problem with discontinuous nonlinearity”, Comput. Math. Math. Phys., 64:6 (2024), 1254–1260
V. N. Pavlenko, D. K. Potapov, “Semiregular solutions of elliptic boundary-value problems with discontinuous nonlinearities of exponential growth”, Sb. Math., 213:7 (2022), 1004–1019
V. N. Pavlenko, D. K. Potapov, “One class of quasilinear elliptic type equations with discontinuous nonlinearities”, Izv. Math., 86:6 (2022), 1162–1178
V. N. Pavlenko, D. K. Potapov, “Positive solutions of superlinear elliptic problems with discontinuous non-linearities”, Izv. Math., 85:2 (2021), 262–278
V. N. Pavlenko, D. K. Potapov, “Variational method for elliptic systems with discontinuous nonlinearities”, Sb. Math., 212:5 (2021), 726–744
V. N. Pavlenko, D. K. Potapov, “Existence of Semiregular Solutions of Elliptic Systems with Discontinuous Nonlinearities”, Math. Notes, 110:2 (2021), 226–241
V. N. Pavlenko, D. K. Potapov, “On a class of elliptic boundary-value problems with parameter and discontinuous non-linearity”, Izv. Math., 84:3 (2020), 592–607
V. N. Pavlenko, D. K. Potapov, “On the existence of three nontrivial solutions of a resonance elliptic boundary value problem with a discontinuous nonlinearity”, Differ. Equ., 56:7 (2020), 831–841
V. N. Pavlenko, D. K. Potapov, “Properties of the spectrum of an elliptic boundary value problem with a parameter and a discontinuous nonlinearity”, Sb. Math., 210:7 (2019), 1043–1066
S. M. Voronin, S. F. Dolbeeva, O. N. Dementev, A. A. Ershov, M. G. Lepchinskii, S. V. Matveev, N. B. Medvedeva, D. K. Potapov, E. A. Rozhdestvenskaya, E. A. Sbrodova, I. M. Sokolinskaya, A. A. Solovev, V. I. Ukhobotov, V. E. Fedorov, “K 70-letiyu professora Vyacheslava Nikolaevicha Pavlenko”, Chelyab. fiz.-matem. zhurn., 2:4 (2017), 383–387
V. N. Pavlenko, D. K. Potapov, “Estimates for a spectral parameter in elliptic boundary value problems with discontinuous nonlinearities”, Siberian Math. J., 58:2 (2017), 288–295
A. M. Kamachkin, D. K. Potapov, V. V. Yevstafyeva, “Existence of solutions for second-order differential equations with discontinuous right-hand side”, Electron. J. Differential Equations, 2016 (2016), 124, 9 pp. http://ejde.math.txstate.edu/Volumes/2016/124/abstr.html
Citation in format AMSBIB
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