Existence and uniqueness of solution for a non-local singular boundary value problem

The main goal of this work is the examination of a mixed singular problem, which is characterized by a singularity of order n at the origin. This problem incorporates a Dirichlet boundary condition alongside an integral with variable bounds. We start by outlining the functional framework for analyzing the posed problem. Initially, we establish a bilateral a priori estimate using energy inequalities. To demonstrate the existence of a solution, we illustrate the density of the image of the operator associated with the problem, relying on regularization operators for our analysis. This approach seeks to demonstrate the existence of a solution to the mixed singular problem, ensuring that the solution adheres to the Dirichlet boundary conditions and the integral with variable bounds. Central to this process are the regularization operators and energy inequalities, which help manage the singularities and establish the existence of a well-defined solution.