Local estimates of renormalized solutions to anisotropic elliptic equations with variable growth and their applications

This paper examines second-order anisotropic elliptic equations with variable nonlinearity exponents and locally summable data. All the main results of the paper are based on new local estimates of a special type obtained by the author in an arbitrary domain Ω of the space ℝⁿ, for both absorption and source equations. Based on these estimates for local renormalized solutions, the following issues are considered: existence and regularity, stability with respect to perturbations of the right-hand side, and extension to sets of small capacity. The obtained results generalize well-known theorems for isotropic cases to equations with anisotropic structure and variable growth.