Sharp and improved regularity estimates for weighted quasilinear elliptic equations of $p-$Laplacian type and applications

In this manuscript, we obtain sharp and improved regularity estimates for weak solutions of weighted quasilinear elliptic models of Hardy-H\'{e}non-type, featuring an explicit regularity exponent depending only on universal parameters. Our approach is based on geometric tangential methods and uses a refined oscillation mechanism, compactness, and scaling techniques. In some specific scenarios, we establish higher regularity estimates and non-degeneracy properties, providing further geometric insights into such solutions. Our regularity estimates both enhance and, to some extent, extend the results arising from the $C^{p^{\prime}}$ conjecture for the $p$-Laplacian with a bounded source term. As applications of our results, we address some Liouville-type results for our class of equations. Finally, our results are noteworthy, even in the simplest model case governed by the $p$-Laplacian with regular coefficients: $$ \mathrm{div}\left( |\nabla u|^{p-2}\mathfrak{A}(|x|) \nabla u\right) = |x|^{\alpha}u_+^m(x) \quad \text{in} \quad B_1 $$ under suitable assumptions on the data, with possibly singular weight $\mathfrak{h}(|x|) = |x|^{\alpha}$, which includes the Matukuma and Batt-Faltenbacher-Horst's equations as toy models.