Boundary value problems for elliptic equations in unbounded domains with conditions at infinity

Mykola Bokalo, Taras Bokalo, Vitaliy Vlasov


DOI: http://dx.doi.org/10.30970/vmm.2025.97.060-078

Анотація


The article investigates a boundary value problems for second-order elli- ptic equations given in unbounded domains. In the class of equations under consideration, in addition to linear ones, there are nonlinear ones with variable nonlinearity exponents.The existence and uniqueness of weak solutions of the studied problems are established under additional conditions on behavior of solutions and the growth of input data at infinity. A priori estimates of weak solutions of the studied problems are obtained. The study uses an analogue of the Saint-Venant principle known from mechanics and the monotonicity method.

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Ivasyshen S.D., Lavrenchuk V.P., Ivasjuk H.P. Reva N.V., Fundamentals of the classical theory of equations of mathematical physics. – Chernivtsi, 2015.

Oleinik O.A., Iosifyan G.A. An analog of Saint-Venant principle and uniqueness of the soluti- ons of the boundary-value problems in unbounded domains for parabolic equations// Usp. Mat. Nauk. – 1976. – V.31, no.6. – P. 142–166.

Shishkov A.E. The solvability of the boundary-value problems for quasilinear elliptic and parabolic equations in unbounded domains in the classes of functions growing at the infini- ty// Ukr. Math. J. – 1985. – V.47, no.2. – P. 277–289.

Patrizia Di Gironimo Existence, regularity, and uniqueness of solutions to some noncoercive nonlinear elliptic equations in unbounded domains// Mathematics. – 2024. – V.12(12), 1860; https://doi.org/10.3390/math12121860.

Br´ezis H. Semilinear equations in RN without conditions at infinity// Appl. Math. Optim.

1984. – V.12, no.3. – P. 271–282.

Bernis F. Elliptic and parabolic semilinear parabolic problems without conditions at infini- ty// Arch. Rational Mech. Anal. – 1989. – V.106, no.3. – P. 217–241.

Bokalo M.M. Boundary value problems for semilinear parabolic equations in unbounded domains without conditions at infinity// Siberian Math. J. – 1996. – V.37, no.5. – P. 860–867.

Bokalo M.M., Buhrii O.M., Hryadil N. Initial-boundary value problems for nonlinear elliptic- parabolic equations with variable exponents of nonlinearity in unbounded domains without conditions at infinity // Nonlinear Analysis. Elsevier. USA. – 2020. – V.192. – P. 1–17.

Ru˙ ˇziˇcka M. Electroreological fluids: modeling and mathematical theory. – Springer-Verl., Berlin, 2000.

Bokalo M., Domanska O. On well-posedness of boundary problems for elliptic equations in general anisotropic Lebesgue-Sobolev spaces// Mat. Stud. – 2007. – V.28, no.1. – P. 77–91.

Bokalo M.M., Domanska O.V. Dirichlet problem for stationary anisotropic higher-ordered partial integrodifferential equations with variable exponents of nonlinearity// Journal of Mathematical Sciences. – 2014. – V.201, no.1. – P. 17–31.

R˘adulescu V., Repovˇs D. Partial differential equations with variable exponents: variational methods and qualitative analysis. – CRC Press, Boca Raton, London, New York, 2015.

Buhrii O., Buhrii N. Nonlocal in time problem for anisotropic parabolic equations with vari- able exponents of nonlinearities// J. Math. Anal. Appl. – 2019. – V.473. – P. 695–711.

Diening L., Harjulehto P., H¨asto¨ P., Ru˙ ˇziˇcka M. Lebesgue and Sobolev spaces with variable exponents. – Springer, Heidelberg, 2011.

Bokalo N.M. Energy estimates for solutions and unique solvability of the Fourier problem for linear and quasilinear parabolic equations// Diff. Equat. – 1994. – V.30, no.8. – P. 1226–1234.

Lions J.-L. Quelques m´ethodes de r´esolution des probl`emes aux limites non lin´eaires. – Paris: Dunod, 1969.


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