On Indirect Approach to the Solvability of Quasi-Linear Dirichlet Elliptic Boundary Value Problem with BMO-Anisotropic p-Laplacian

• Main Authors: Peter I. Kogut, Olha P. Kupenko
• Format: Article
• Language: English
• Published: Oles Honchar Dnipro National University 2018-12-01
• Series: Journal of Optimization, Differential Equations and Their Applications
• Subjects: Anisotropic p-Laplacian; approximation procedure; weak solutions; BMO-coefficients
• Online Access: https://model-dnu.dp.ua/index.php/SM/article/view/129

We study here Dirichlet boundary value problem for a quasi-linear elliptic equation with anisotropic p-Laplace operator in its principle part and L^1-control in coefficient of the low-order term. As characteristic feature of such problem  is a specification of the matrix of anisotropy A=A^{sym}+A^{skew} in BMO-space. Since we cannot expect to have a solution of the state equation in the classical Sobolev space W^{1,p}_0(\Omega), we specify a suitable functional class in which we look for solutions and prove existence of weak solutions in the sense of Minty using a non standard approximation procedure and compactness arguments in variable spaces.