The main result establishes that a weak solution of degenerate elliptic equations can be approximated by a sequence of solutions of non-degenerate elliptic equations. To this end we prove an approximation theorem for Ap-weights.

In this paper, we survey a number of recent results obtained in the study of weighted Sobolev spaces (with power-type weights, Ap-weights, padmissible weights, regular weights and the conjecture of De Giorgi) and the existence of entropy solutions for degenerate quasilinear elliptic equations.

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