1683-3414 1814-0807 Владикавказский математический журнал Vladikavkaz Mathematical Journal Южный математический институт - филиал Федерального государственного бюджетного учреждения науки Федерального научного центра «Владикавказский научный центр Российской академии наук» (ЮМИ ВНЦ РАН) Existence and Uniqueness of Solution for Nonlinear Anisotropic Elliptic Dirichlet Problems Существование и единственность решения для нелинейных анизотропных эллиптических задач Дирихле 10.46698/c8515-1572-8469-r 18591 03 2026 28 1 122 133 https://vmj.ru/eng/archive/detail.php?ELEMENT_ID=18627&SECTION_ID=658 Насери М. nasrimokhtar@gmail.com, m.naceri@ens-lagh.dz Высшая педагогическая школа Лагуата, Алжир, 03000, Лагуат, пр. Мартирс , We consider boundary value problem for nonlinear anisotropic elliptic partial differential equations in bounded open Lipschitz domain and the Dirichlet boundary conditions. We also suppose that the body force function belongs to the natural dual space under certain hypotheses regarding the nonlinear anisotropic operators present on the main side of the proposed problems. We prove the existence and uniqueness of a weak solution in anisotropic Sobolev space for this problem. Our proofs are based on various anisotropic Sobolev inequalities, embedding theorems, and features of pseudo-monotone operators. The functional setting involves anisotropic Lebesgue and Sobolev spaces in the scalar case and their most important properties. Рассматривается краевая задача для нелинейных анизотропных эллиптических дифференциальных уравнений в частных производных в ограниченной открытой липшицевой области и граничными условиями Дирихле. При этом предполагается, что функция внешних сил принадлежит естественному двойственному пространству при определенных гипотезах относительно нелинейных анизотропных операторов, присутствующих в основной части предлагаемых задач. Центральный результат представляет собой доказательство существования и единственности слабого решения в анизотропном пространстве Соболева для этой задачи. Оно основывается на применении различных анизотропных неравенств Соболева, теорем вложения и определенных особенностях псевдомонотонных операторов. Отметим, что функциональная постановка задачи включает анизотропные пространства Лебега и Соболева в скалярном случае и их наиболее важные свойства. нелинейные эллиптические уравнения анизотропные пространства Соболева слабое решение существование единственность nonlinear elliptic equations anisotropic Sobolev spaces weak solution existence uniqueness. 1 Ruzicka, M. Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, vol. 1748, Springer, Berlin, 2000. Ruzicka, M. Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, vol. 1748, Springer, Berlin, 2000. 2 Chen, Y., Levine, S. and Rao, M. 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