On the existence of solutions for degenerate elliptic problems with singular nonlinearities

This paper investigates the existence of nonnegative weak solutions to a class of degenerate elliptic equations with singular nonlinearities. The problem under consideration is of the form Formula: see text with homogeneous Dirichlet boundary conditions, where Formula: see text is a bounded domain, Formula: see text, Formula: see text, Formula: see text is a nonnegative element of the dual Sobolev space Formula: see text, and Formula: see text is a continuous function that may blow up at zero but remains bounded at infinity. The degeneracy of the principal part, controlled by the parameter Formula: see text, adds significant difficulty to the analysis. Using a double approximation scheme (regularizing both the degeneracy and the singularity), truncation arguments, monotonicity methods, and the Schauder fixed point theorem, we establish the existence of a solution Formula: see text under appropriate conditions on the data. Our main contribution lies in the simultaneous treatment of degeneracy and singularity, extending classical results to a broader class of non-uniformly elliptic operators. The proofs rely on uniform a priori estimates, compactness arguments, and a careful passage to the limit in the approximate problems.