## Abstract

We consider the question of the existence of the Dirichlet problem for second-order elliptic equations with spectral parameter and a nonlinearity discontinuous with respect to the phase variable. Here it is not assumed that the differential operator is formally selfadjoint. Using the method of upper and lower solutions, we establish results on the existence of nontrivial (positive and negative) solutions under positive values of the spectral parameter for the problems under study.

### Similar content being viewed by others

## References

H. Amann, “On the number of solutions of nonlinear equations in ordered Banach spaces,” J. Funct. Anal.

**11**, 346 (1972).H. Amann, “Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,” SIAM Rev.

**18**, 620 (1976).H. Amann, “Existence and multiplicity theorems for semi-linear elliptic boundary value problems,” Math. Z.

**150**, 281 (1976).H. Amann, “Supersolutions, monotone iterations, and stability,” J. Differential Equations

**21**, 363 (1976).H. Amann and M. G. Crandall, “On some existence theorems for semi-linear elliptic equations,” Indiana Univ. Math. J.

**27**, 779 (1978).N. Basile and M. Mininni, “Some solvability results for elliptic boundary value problems in resonance at the first eigenvalue with discontinuous nonlinearities,” Boll. Un. Mat. Ital. B (5)

**17**, 1023 (1980).G. Bonanno G. and P. Candito, “Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities,” J. Differential Equations

**244**, 3031 (2008).K. C. Chang, “The obstacle problem and partial differential equations with discontinuous nonlinearities,” Comm. Pure Appl. Math.

**33**, 117 (1980).H. Chrayteh and J. M. Rakotoson, “Eigenvalue problems with fully discontinuous operators and critical exponents,” Nonlinear Anal.

**73**, 2036 (2010).R. Iannacci, M. N. Nkashama and J. R. Ward, “Nonlinear second order elliptic partial differential equations at resonance,” Trans. Amer. Math. Soc.

**311**, 711 (1989).M. A. Krasnosel’skiĭand A. V. Pokrovskiĭ, “Regular solutions to elliptic equations with discontinuous nonlinearities,”

*Proceedings of the All-Union Conference on Partial Differential Equations Dedicated to the 75th Anniversary of Academician I. G. Petrovskii*(Izd-vo Mosk. Univ., Moscow, 1978), 346.M. A. Krasnosel’skiĭand A. V. Pokrovskiĭ, “Elliptic equations with discontinuous nonlinearities,” Dokl. Akad. Nauk 342, 731–734 (1995) [Dokl. Math.

**51**, 415–418 (1995)].M. A. Krasnosel’skiĭand A. V. Sobolev “Fixed points of discontinuous operators,” Sib. Mat. Zh.

**14**, 674 (1973) [Sib. Math. J.**14**, 470 (1973)].O. A. Ladyzhenskaya and N. N. Ural’tseva,

*Linear and Quasilinear Elliptic Equations*(Nauka, Moscow, 1964; Academic Press, New York–London, 1968).S. A. Marano and D. Motreanu, “On a three critical points theorem for non-differentiable functions and applications to nonlinear boundary value problems,” Nonlinear Anal.

**48**, 37 (2002).I. Massabo, “Elliptic boundary value problems at resonance with discontinuous nonlinearities,” Boll. Un. Mat. Ital. B (5)

**17**, 1308 (1980).V. N. Pavlenko, “Existence of semiregular solutions of a first boundary-value problem for a parabolic equation with a nonmonotonic discontinuous nonlinearity,” Differ. Uravn.

**27**, 520 (1991) [Differ. Equ.**27**, 374 (1991)].V. N. Pavlenko and D. K. Potapov, “Existence of a ray of eigenvalues for equations with discontinuous operators,” Sib. Mat. Zh.

**42**, 911 (2001) [Sib. Math. J.**42**, 766 (2001)].V. N. Pavlenko and O. V. Ul’ yanova, “The method of upper and lower solutions for equations of elliptic type with discontinuous nonlinearities,” Izv. Vyssh. Uchebn. Zaved., Mat. 1998, No.

**11**, 69 (1998) [Russ. Math.**42**(11), 65 (1998)].V. N. Pavlenko and O. V. Ul’yanova, “Method of upper and lower solutions for parabolic-type equations with discontinuous nonlinearities,” Differ. Uravn.

**38**, 499 (2002) [Differ. Equ.**38**, 520 (2002)].D. K. Potapov, “On the existence of a ray of eigenvalues for equations of elliptic type with discontinuous nonlinearities in the critical case,” Vestnik SPbGU Ser. 10. Applied Mathematics. Computer Science. Control Processes. Issue.

**4**, 125 (2004).D. K. Potapov, “On an upper bound for the value of the bifurcation parameter in eigenvalue problems for elliptic equations with discontinuous nonlinearities,” Differ. Uravn.

**44**, 715 (2008) [Differ. Equ.**44**, 737 (2008)].D. K. Potapov, “Continuous approximations of Gol’dshtik’s model,” Mat. Zametki

**87**, 262 (2010) [Math. Notes**87**, 244 (2010); Erratum Math. Notes**87**, 453 ({2010)].D. K. Potapov, “On the eigenvalue set structure for higher-order equations of elliptic type with discontinuous nonlinearities,” Differ. Uravn.

**46**, 150 (2010) [Differ. Equ.**46**, 155 (2010)].D. K. Potapov, “On a “separating” set for higher-order equations of elliptic type with discontinuous nonlinearities,” Differ. Uravn.

**46**, 451 (2010) [Differ. Equ.**46**, 458 (2010)].D. K. Potapov, “Bifurcation problems for equations of elliptic type with discontinuous nonlinearities,” Mat. Zametki

**90**, 280 (2011) [Math. Notes**90**, 260 (2011)].D. K. Potapov, “On solutions of the Gol’dshtik problem,” Sib. Zh. Vychisl. Mat.

**15**, 409 (2012) [Numer. Analysis Appl.**5**, 342 (2012)].D. K. Potapov, “On the number of semiregular solutions in problems with spectral parameter for higher-order equations of elliptic type with discontinuous nonlinearities,” Differ. Uravn.

**48**, 447 (2012) [Differ. Equ.**48**, 455 (2012)].D. K. Potapov, “On one problem of electrophysics with discontinuous nonlinearity,” Differ. Uravn.

**50**, 421 (2012) [Differ. Equ.**50**, 419 (2012)].D. K. Potapov and V. V. Yevstafyeva, “Lavrent’ev problem for separated flows with an external perturbation,” Electron. J. Differential Equations. No.

**255**, 1 (2013).K. Schmitt, “Revisiting the method of sub- and supersolutions for nonlinear elliptic problems,” Electron. J. Differential Equations. No. Conf.

**15**, 377 (2007).C. A. Stuart, “Maximal and minimal solutions of elliptic differential equations with discontinuous nonlinearities,” Math. Z.

**163**, 239 (1978).C. A. Stuart and J. F. Toland, “A property of solutions of elliptic differential equations with discontinuous nonlinearities,” J. London Math. Soc. (2)

**21**, 329 (1980).C. Wang and Y. Huang, “Multiple solutions for a class of quasilinear elliptic problems with discontinuous nonlinearities and weights,” Nonlinear Anal.

**72**, 4076 (2010).

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

Original Russian Text © V.N. Pavlenko and D.K. Potapov, 2016, published in Matematicheskie Trudy, 2016, Vol. 19, No. 1, pp. 91–105.

## About this article

### Cite this article

Pavlenko, V.N., Potapov, D.K. Existence of solutions to a nonvariational elliptic boundary value problem with parameter and discontinuous nonlinearity.
*Sib. Adv. Math.* **27**, 16–25 (2017). https://doi.org/10.3103/S1055134417010023

Received:

Published:

Issue Date:

DOI: https://doi.org/10.3103/S1055134417010023