Geodesic
Positive solutions of superlinear elliptic~problems with discontinuous non-linearities
Izvestiya. Mathematics , Tome 85 (2021) no. 2, pp. 262-278.

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We consider an elliptic boundary-value problem with a homogeneous Dirichlet boundary condition, a parameter and a discontinuous non-linearity. The positive parameter appears as a multiplicative term in the non-linearity, and the problem has a zero solution for any value of the parameter. The non-linearity has superlinear growth at infinity. We prove the existence of positive solutions by a topological method.
Keywords: superlinear elliptic problem, parameter, discontinuous non-linearity, topological method.
Mots-clés : positive solution 
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V. N. Pavlenko; D. K. Potapov. Positive solutions of superlinear elliptic~problems with discontinuous non-linearities. Izvestiya. Mathematics , Tome 85 (2021) no. 2, pp. 262-278. http://geodesic.mathdoc.fr/item/IM2_2021_85_2_a3/

[1] 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 | DOI | DOI | MR | Zbl

[2] V. N. Pavlenko, D. K. Potapov, “Existence of solutions to a nonvariational elliptic boundary value problem with parameter and discontinuous nonlinearity”, Siberian Adv. Math., 27:1 (2017), 16–25 | DOI | DOI | MR | Zbl

[3] G. Barletta, A. Chinnì, D. O'Regan, “Existence results for a Neumann problem involving the $p(x)$-Laplacian with discontinuous nonlinearities”, Nonlinear Anal. Real World Appl., 27 (2016), 312–325 | DOI | MR | Zbl

[4] S. Bensid, “Perturbation of the free boundary in elliptic problem with discontinuities”, Electron. J. Differential Equations, 2016 (2016), 132, 14 pp. | MR | Zbl

[5] R. Dhanya, S. Prashanth, S. Tiwari, K. Sreenadh, “Elliptic problems in $\mathbb{R}^N$ with critical and singular discontinuous nonlinearities”, Complex Var. Elliptic Equ., 61:12 (2016), 1656–1676 | DOI | MR | Zbl

[6] V. N. Pavlenko, D. K. Potapov, “Existence of two nontrivial solutions for sufficiently large values of the spectral parameter in eigenvalue problems for equations with discontinuous right-hand sides”, Sb. Math., 208:1 (2017), 157–172 | DOI | DOI | MR | Zbl

[7] V. N. Pavlenko, D. K. Potapov, “Existence of three nontrivial solutions of an elliptic boundary-value problem with discontinuous nonlinearity in the case of strong resonance”, Math. Notes, 101:2 (2017), 284–296 | DOI | DOI | MR | Zbl

[8] 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 | DOI | DOI | MR | Zbl

[9] G. C. G. dos Santos, G. M. Figueiredo, “Existence of solutions for an NSE with discontinuous nonlinearity”, J. Fixed Point Theory Appl., 19:1 (2017), 917–937 | DOI | MR | Zbl

[10] S. Heidarkhani, F. Gharehgazlouei, “Multiplicity of elliptic equations involving the $p$-Laplacian with discontinuous nonlinearities”, Complex Var. Elliptic Equ., 62:3 (2017), 413–429 | DOI | MR | Zbl

[11] 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 | DOI | DOI | MR | Zbl

[12] 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 | DOI | DOI | MR | Zbl

[13] D. K. Potapov, “On solutions to the Goldshtik problem”, Numer. Anal. Appl., 5:4 (2012), 342–347 | DOI | Zbl

[14] D. K. Potapov, V. V. Yevstafyeva, “Lavrent'ev problem for separated flows with an external perturbation”, Electron. J. Differential Equations, 2013 (2013), 255, 6 pp. | MR | Zbl

[15] Y. Zhang, I. Danaila, “Existence and numerical modelling of vortex rings with elliptic boundaries”, Appl. Math. Model., 37:7 (2013), 4809–4824 | DOI | MR | Zbl

[16] D. K. Potapov, “On one problem of electrophysics with discontinuous nonlinearity”, Differ. Equ., 50:3 (2014), 419–422 | DOI | DOI | MR | Zbl

[17] V. N. Pavlenko, D. K. Potapov, “Elenbaas problem of electric arc discharge”, Math. Notes, 103:1 (2018), 89–95 | DOI | DOI | MR | Zbl

[18] M. A. Krasnosel'skiĭ, A. V. Pokrovskiĭ, Systems with hysteresis, Springer-Verlag, Berlin, 1989, xviii+410 pp. | DOI | MR | MR | Zbl | Zbl

[19] D. K. Potapov, “On the eigenvalue set structure for higher-order equations of elliptic type with discontinuous nonlinearities”, Differ. Equ., 46:1 (2010), 155–157 | DOI | MR | Zbl

[20] I. V. Shragin, “Conditions for measurability of superpositions”, Soviet Math. Dokl., 12 (1971), 465–470 | MR | Zbl

[21] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren Math. Wiss., 224, 2nd ed., Springer-Verlag, Berlin, 1983, xiii+513 pp. | DOI | MR | MR | Zbl | Zbl

[22] C. A. Stuart, “Maximal and minimal solutions of elliptic differential equations with discontinuous non-linearities”, Math. Z., 163:3 (1978), 239–249 | DOI | MR | Zbl

[23] Kung-Ching Chang, “Variational methods for non-differentiable functionals and their applications to partial differential equations”, J. Math. Anal. Appl., 80:1 (1981), 102–129 | DOI | MR | Zbl

[24] Yu. G. Borisovich, B. D. Gelman, A. D. Myshkis, V. V. Obukhovskii, Vvedenie v teoriyu mnogoznachnykh otobrazhenii i differentsialnykh vklyuchenii, 2-e izd., ispr. i dop., Librokom, M., 2011, 224 pp. | MR | Zbl

[25] Tsoy-wo Ma, Topological degrees of set-valued compact fields in locally convex spaces, Dissertationes Math. (Rozprawy Mat.), 92, Instytut Matematyczny Polskiej Akademii Nauk, Warszawa, 1972, 43 pp. | MR | Zbl

[26] H. J. Kuiper, “On positive solutions of nonlinear elliptic eigenvalue problems”, Rend. Circ. Mat. Palermo (2), 20:2-3 (1971), 113–138 | DOI | MR | Zbl

[27] H. Brezis, R. E. L. Turner, “On a class of superlinear elliptic problems”, Comm. Partial Differential Equations, 2:6 (1977), 601–614 | DOI | MR | Zbl

[28] W. Allegretto, P. Nistri, “Elliptic equations with discontinuous nonlinearities”, Topol. Methods Nonlinear Anal., 2:2 (1993), 233–251 | DOI | MR | Zbl

[29] M. A. Krasnosel'skiĭ, Positive solutions of operator equations, P. Noordhoff Ltd., Groningen, 1964, 381 pp. | MR | MR | Zbl | Zbl

[30] R. Courant, D. Hilbert, Methods of mathematical physics, v. II, Partial differential equations, (vol. II by R. Courant), Interscience Publishers, a division of John Wiley Sons, New York–London, 1962, xxii+830 pp. | MR | MR | Zbl | Zbl