Existence and regularity of solutions of nonlinear anisotropic elliptic problem with Hardy potential.

This article investigates the existence and regularity of solutions to a class of nonlinear anisotropic elliptic equations involving a Hardy potential and data in Lebesgue spaces \(L^m(\Omega)\), within appropriate anisotropic Sobolev spaces. The main focus is on establishing natural conditions related to an anisotropic Hardy inequality that guarantee the existence and summability of weak solutions, even when the right-hand side includes singular terms (Hardy potentials). The work extends previous isotropic results to the anisotropic setting, addressing challenges such as singularities at the origin and the application of anisotropic Hardy inequalities. Key results include a priori estimates for approximate solutions and convergence arguments that ensure the existence of weak solutions under specified integrability and parameter constraints. The paper also notes that these results hold for more general nonlinear operators satisfying certain Carathéodory conditions.