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

Shaya Shakerian, Jérôme Vétois

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.

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

Paper Structure

This paper contains 5 sections, 2 theorems, 135 equations.

Table of Contents

  1. Introduction and main results
  2. Regularity and boundedness results
  3. The upper bound estimates
  4. The first-order term
  5. The Kelvin-type transformation

Key Result

Theorem 1.1

Let $n$, $p$, $a$, $b$ and $q$ be such that K-C-N ineq:conditions holds true, $f:\mathbb{R}^n\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function satisfying Th1Eq1 and $u$ be a solution of Main Problem. Then there exists a constant $C_0>0$ such that where $\mu:=$n-p$1+a$$/$p-1$$. If moreover $u>0$ and $f$x,u$\ge0$ in $\mathbb{R}^n$, then there exist constants $\alpha,\delta,C_1>0$ such that

Theorems & Definitions (7)

  • Theorem 1.1
  • Proposition 1.1
  • Definition 1.1
  • Remark 1.1
  • Definition 2.1
  • Definition 2.2
  • proof