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

Source abstract

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.