Results on entire solutions for a degenerate critical elliptic equation with anisotropic coefficients

TL;DR

This paper investigates entire solutions of a degenerate critical elliptic equation with anisotropic coefficients, establishing properties, regularity, symmetry, uniqueness, and nonexistence results.

Contribution

It provides new insights into the properties and solutions of degenerate anisotropic elliptic equations, including regularity, symmetry, and nonexistence results.

Findings

Properties of the degenerate elliptic operator are characterized.

Regularity, symmetry, and uniqueness of solutions are established.

Nonexistence results for certain solutions are derived.

Abstract

In this paper, we study the following degenerate critical elliptic equations with anisotropic coefficients $$ -div(|x_{N}|^{2\alpha}\nabla u)=K(x)|x_{N}|^{\alpha\cdot 2^{*}(s)-s}|u|^{2^{*}(s)-2}u {in} \mathbb{R}^{N} $$ where $x=(x_{1},...,x_{N})\in\mathbb{R}^{N},$ $N\geq 3,$ $\alpha>1/2,$ $0\leq s\leq 2$ and $2^{*}(s)=2(N-s)/(N-2).$ Some basic properties of the degenerate elliptic operator $-div(|x_{N}|^{2\alpha}\nabla u)$ are investigated and some regularity, symmetry and uniqueness results for entire solutions of this equation are obtained. We also get some variational identities for solutions of this equation. As a consequence, we obtain some nonexistence results for solutions of this equation.