ENTROPY SOLUTIONS FOR P-LAPLACIAN TYPE PROBLEMS WITH LOWER-ORDER PERTURBATION AND SINGULAR TERMS IN WEIGHTED FRAMEWORK

Abstract

Abstract. This paper is devoted to the study of a class of nonlinear elliptic problems involving weighted degenerate operators of p-Laplacian type, lower-order perturbations, and singular source terms. Such problems arise in the mathematical modeling of heterogeneous media and diffusion processes characterized by nonuniform physical properties, where degeneracy and singular behavior significantly complicate the analysis. The main objective of the work is to establish the existence of entropy solutions for a broad family of weighted elliptic equations with merely integrable data. The analysis is carried out within the framework of weighted Sobolev spaces associated with Muckenhoupt weights, which provide a suitable setting for handling both degeneracy and lack of regularity. The methodology relies on the construction of appropriate approximating problems, truncation techniques, uniform a priori estimates, compactness arguments, and convergence methods adapted to entropy solutions. Under suitable assumptions on the weight function, the nonlinear perturbation term, and the singular nonlinearity, the existence of at least one entropy solution is proved. The obtained results extend several earlier existence theorems by simultaneously incorporating weighted degeneracy, lower order perturbations, and singular reaction terms into a unified framework. This study contributes to the theory of nonlinear elliptic partial differential equations by enlarging the class of problems for which solvability can be guaranteed with low regularity data. Moreover, the developed approach provides useful analytical tools for future investigations of weighted singular problems arising in mathematical physics, engineering, and related applied sciences.
Keywords: Entropy solutions, Existence results, Singular non-linearity, Weighted Sobolev spaces.