Strong \(A_\infty\) weights are introduced and degenerate elliptic equations with respect to strong weights are then studied. We prove Harnack inequality and local regularity results for weak solutions of a quasilinear degenerate equation in divergence form under natural growth conditions. The degeneracy is given by a suitable power of a strong \(A_\infty\) weight. Regularity results are achieved under minimal assumptions on the coefficients. Then we point two applications. First we prove \(C^{1,\alpha}\) local estimates for solutions of a degenerate equation in non divergence form. As a second application we prove a unique continuation property for positive weak solutions of degenerate elliptic equations.

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