Sharp pointwise estimates for weighted critical $p$-Laplace equations

TL;DR

This paper derives sharp pointwise estimates for solutions to weighted critical p-Laplace equations, extending previous unweighted results and refining asymptotic behavior analysis using Kelvin-type transformations.

Contribution

It introduces new sharp pointwise estimates for weighted p-Laplace equations and refines asymptotic analysis via Kelvin-type transformations in the quasilinear setting.

Findings

Extended sharp estimates to weighted cases.

Refined asymptotic expansion at infinity.

Applied Kelvin-type transformation to quasilinear equations.

Abstract

We investigate the asymptotic behavior of solutions to a class of weighted quasilinear elliptic equations which arise from the Euler--Lagrange equation associated with the Caffarelli--Kohn--Nirenberg inequality. We obtain sharp pointwise estimates which extend and improve previous results obtained in the unweighted case. In particular, we show that we can refine the asymptotic expansion at infinity by using a Kelvin-type transformation, which reduces the problem to another elliptic-type problem near the origin. The application of this transformation is straightforward in the linear case but more delicate in the quasilinear case. In particular, it is necessary in this case to establish some preliminary estimates before being able to apply the transformation.