Abstract
We consider the Dirichlet problem for an equation of the elliptic type with a nonlinearity discontinuous with respect to the phase variable in the resonance case; it is not required that the nonlinearity satisfy the Landesman-Lazer condition. Using the regularization of the original equation, we establish the existence of a generalized solution of the problem indicated.
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REFERENCES
M. A. Krasnosel’skii and A. V. Pokrovskii, Systems with Hysteresis [in Russian], Nauka, Moscow (1983).
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin (1983).
E. Landesman and A. Lazer, “Nonlinear perturbations of linear elliptic boundary value problems at resonance,” J. Math. Mech., 19, No.7, 609–623 (1970).
N. Basile and M. Mininni, “Some solvability results for elliptic boundary value problems in resonance at the first eigenvalue with discontinuous nonlinearities,” Boll. Unione Math. Ital., 117-B, No.3, 1023–1033 (1980).
I. Massabo, “Elliptic boundary value problems at resonance with discontinuous nonlinearities,” Boll. Unione Math. Ital., Ser. 5, 17-B, No.3, 1302–1320 (1980).
K.-C. Chang, “Variational methods for nondifferentiable function and their applications to partial differential equations,” J. Math. Anal. Appl., 80, No.1, 102–129 (1981).
V. N. Pavlenko and V. V. Vinokur, “Resonance boundary-value problems for elliptic equations with discontinuous nonlinearity,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 5, 43–58 (2001).
V. N. Pavlenko and V. V. Vinokur, “Existence theorems for equations with noncoercive discontinuous operators,” Ukr. Mat. Zh., 54, No.3, 349–363 (2002).
P. Bartolo, V. Benci, and D. Fortunato, “Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity,” Nonlin. Anal., 7, No.9, 981–1012 (1983).
R. Iannacci, M. N. Nkashama, and J. R. Ward, “Nonlinear second order elliptic partial differential equations at resonance,” Proc. Amer. Math. Soc., 311, No.2, 711–725 (1989).
V. N. Pavlenko and E. A. Chizh, “Dirichlet problem for the Laplace equation with discontinuous nonlinearity and without Landesman-Lazer condition,” Vestn. Chelyabinsk. Univ., Ser. 3, Mat. Mekh. Inform., No. 1 (16), 120–126 (2002).
O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations [in Russian], Nauka, Moscow (1964).
V. N. Pavlenko, “Control over singular distributed systems of parabolic type with discontinuous nonlinearities,” Ukr. Mat. Zh., 46, No.6, 729–736 (1994).
T. W. Ma, “Topological degree for set valued compact vector fields in locally convex spaces,” Rozpawy Mat., 92, 3–47 (1972).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 1, pp. 102–110, January, 2005.
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Pavlenko, V.N., Chizh, E.A. Existence Theorem for One Class of Strongly Resonance Boundary-Value Problems of Elliptic Type with Discontinuous Nonlinearities. Ukr Math J 57, 121–131 (2005). https://doi.org/10.1007/s11253-005-0175-x
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DOI: https://doi.org/10.1007/s11253-005-0175-x