Abstract
Sufficient conditions for the existence of a generalized solution to a nonlinear elliptic differential equation with nonlocal boundary conditions of Bitsadze–Samarskii type are proved. The strong ellipticity condition is used for an auxiliary differential-difference operator. Under the formulated conditions, the differential-difference operator is demicontinuous, coercive, and has a semibounded variation, so the general theory of pseudomonotone operators can be applied.
This is a preview of subscription content, log in via an institution to check access.
Notes
In the definitions, we consider abstract Banach spaces \(X\) and \(Y\).
For now we assume that \(\rho < p - 1\). The situation \(\rho = p - 1\) will be explained at the end of the proof.
REFERENCES
A. V. Bitsadze and A. A. Samarskii, “On some simple generalized linear elliptic problems,” Dokl. Akad. Nauk SSSR 185 (4), 739–740 (1969).
A. A. Samarskii, “On some problems in the theory of differential equations,” Differ. Uravn. 16 (1), 1925–1935 (1980).
A. L. Skubachevskii, “Nonlocal elliptic problems with a parameter,” Math. USSR-Sb. 49 (1), 197–206 (1984).
A. L. Skubachevskii, “Elliptic problems with nonlocal conditions near the boundary,” Math. USSR-Sb. 57 (1), 293–316 (1987).
A. L. Skubachevskii, Elliptic Functional Differential Equations and Applications (Birkhäuser, Basel, 1997).
A. L. Skubachevskii, “Nonclassical boundary value problems I,” J. Math. Sci. 155 (2), 199–334 (2008).
A. L. Skubachevskii, “Nonclassical boundary value problems II,” J. Math. Sci. 166 (4), 377–561 (2010).
O. V. Solonukha, “On a nonlinear nonlocal problem of elliptic type,” Comput. Math. Math. Phys. 57 (3), 422–433 (2017).
O. V. Solonukha, “Generalized solutions of quasilinear elliptic differential-difference equations,” Comput. Math. Math. Phys. 60 (12), 2019–2031 (2020).
O. V. Solonukha, “Existence of solutions of parabolic variational inequalities with one-sided restrictions,” Math. Notes 77 (3), 424–439 (2005).
O. V. Solonukha, “On a class of essentially nonlinear elliptic differential-difference equations,” Proc. Steklov Inst. Math. 283, 226–244 (2013).
O. V. Solonukha, “On nonlinear and quasilinear elliptic functional-differential equations,” Discrete Continuous Dyn. Sys. Ser. S 9 (3), 847–868 (2016).
M. A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations (Gostekhizdat, Moscow, 1956; Pergamon, New York, 1964).
S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics (Am. Math. Soc., Providence, RI, 1963).
Yu. A. Dubinskii, “Nonlinear elliptic and parabolic equations,” Advances in Science and Technology: Modern Problems in Mathematics (VINITI, Moscow, 1976), Vol. 9, pp. 5–130 [in Russian].
V. I. Ivanenko and V. S. Mel’nik, Variational Methods in Control Problems for Systems with Distributed Parameters (Naukova Dumka, Kiev, 1988) [in Russian].
Funding
This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The author of this work declares that she has no conflicts of interest.
Additional information
Translated by I. Ruzanova
Publisher’s Note.
Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Solonukha, O.V. On the Solvability of an Essentially Nonlinear Elliptic Differential Equation with Nonlocal Boundary Conditions. Comput. Math. and Math. Phys. 64, 285–299 (2024). https://doi.org/10.1134/S096554252402012X
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S096554252402012X