Regularity estimates for weighted quasilinear elliptic models of p-Laplacian type.

In this Lecture, we obtain sharp and improved regularity estimates for weak solutions of weighted quasilinear elliptic models of Hardy-Hénon-type, featuring an explicit regularity exponent depending only on universal parameters. We also establish higher regularity estimates and non-degeneracy properties in some specific scenarios, 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. Finally, our results are noteworthy, even in the simplest model case governed by the p-Laplacian with regular coefficients, under suitable assumptions on the data, with possibly singular weights, which includes the Matukuma and Batt–Faltenbacher–Horst's equations as toy models.

This is a joint work with Disson dos Prazeres (Universidade Federal de Sergipe - Brazil), Gleydson C. Ricarte (Universidade Federal do Ceará - Brazil), and Ginaldo S. Sá (Universidad de Chile - Chile).