Weak semiregular solutions to the Dirichlet problem for quasilinear elliptic equations in divergence form with discontinuous weak nonlinearities

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Weak semiregular solutions to the Dirichlet problem for quasilinear elliptic equations in divergence form with discontinuous weak nonlinearities. / Pavlenko, V.N.; Potapov, D.K.

In: Sbornik Mathematics, Vol. 216, No. 6, 2025, p. 807-821.

Research output: Contribution to journal › Article › peer-review

BibTeX

@article{d5211f853fdb48e4a68f57ea2763e070,

title = "Weak semiregular solutions to the Dirichlet problem for quasilinear elliptic equations in divergence form with discontinuous weak nonlinearities",

abstract = "In a bounded domain of an N-dimensional space we study the homogeneous Dirichlet problem for a quasilinear elliptic equation in divergence form with a discontinuous weak nonlinearity of power growth at infinity. Using a variational method based on the concept of quasipotential operator we obtain a theorem on the existence of a weak semiregular solution to the problem under study. The semiregularity of the solution means that, almost everywhere in the domain in which the boundary value problem is considered, its values are continuity points of the weak nonlinearity with respect to the phase variable. Next, a positive parameter is introduced into the equation as a multiplier of the weak nonlinearity, and the question of the existence of nontrivial weak semiregular solutions to the resulting boundary value problem is studied. In this case the existence of a trivial solution for all values of the parameter is assumed. A theorem on the existence of a nontrivial weak semiregular solution for sufficiently large values of the parameter is established.",

keywords = "discontinuous weak nonlinearity, equation of nonlinear stationary filtration, quasipotential operator, variational method, weak semiregular solution",

author = "V.N. Pavlenko and D.K. Potapov",

year = "2025",

doi = "10.4213/sm10166e",

language = "English",

volume = "216",

pages = "807--821",

journal = "Sbornik Mathematics",

issn = "1064-5616",

publisher = "Turpion Ltd.",

number = "6",

}

RIS

TY - JOUR

T1 - Weak semiregular solutions to the Dirichlet problem for quasilinear elliptic equations in divergence form with discontinuous weak nonlinearities

AU - Pavlenko, V.N.

AU - Potapov, D.K.

PY - 2025

Y1 - 2025

N2 - In a bounded domain of an N-dimensional space we study the homogeneous Dirichlet problem for a quasilinear elliptic equation in divergence form with a discontinuous weak nonlinearity of power growth at infinity. Using a variational method based on the concept of quasipotential operator we obtain a theorem on the existence of a weak semiregular solution to the problem under study. The semiregularity of the solution means that, almost everywhere in the domain in which the boundary value problem is considered, its values are continuity points of the weak nonlinearity with respect to the phase variable. Next, a positive parameter is introduced into the equation as a multiplier of the weak nonlinearity, and the question of the existence of nontrivial weak semiregular solutions to the resulting boundary value problem is studied. In this case the existence of a trivial solution for all values of the parameter is assumed. A theorem on the existence of a nontrivial weak semiregular solution for sufficiently large values of the parameter is established.

AB - In a bounded domain of an N-dimensional space we study the homogeneous Dirichlet problem for a quasilinear elliptic equation in divergence form with a discontinuous weak nonlinearity of power growth at infinity. Using a variational method based on the concept of quasipotential operator we obtain a theorem on the existence of a weak semiregular solution to the problem under study. The semiregularity of the solution means that, almost everywhere in the domain in which the boundary value problem is considered, its values are continuity points of the weak nonlinearity with respect to the phase variable. Next, a positive parameter is introduced into the equation as a multiplier of the weak nonlinearity, and the question of the existence of nontrivial weak semiregular solutions to the resulting boundary value problem is studied. In this case the existence of a trivial solution for all values of the parameter is assumed. A theorem on the existence of a nontrivial weak semiregular solution for sufficiently large values of the parameter is established.

KW - discontinuous weak nonlinearity

KW - equation of nonlinear stationary filtration

KW - quasipotential operator

KW - variational method

KW - weak semiregular solution

UR - https://www.mendeley.com/catalogue/4f52bd3f-d820-3d14-bfdf-5aed6a83ef4e/

U2 - 10.4213/sm10166e

DO - 10.4213/sm10166e

M3 - Article

VL - 216

SP - 807

EP - 821

JO - Sbornik Mathematics

JF - Sbornik Mathematics

SN - 1064-5616

IS - 6

ER -

ID: 139442450