A note on the classification of positive solutions to the critical p-Laplace equation in R n Rⁿ

In this note, we obtain a classification result for positive solutions to the critical p -Laplace equation in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> Rⁿ with n ≥ 4 and p > p n for some number <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>∈</m:mo> <m:mfenced close=")" open="("> <m:mrow> <m:mfrac> <m:mrow> <m:mi>n</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:mfrac> <m:mo>,</m:mo> <m:mfrac> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:mfrac> </m:mrow> </m:mfenced> </m:math> pₙ∈ (n/3,n+1/3) such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>∼</m:mo> <m:mfrac> <m:mrow> <m:mi>n</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:mfrac> <m:mo>+</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:mfrac> </m:math> pₙ~ n/3+1/n , which improves upon a similar result obtained by Ou (“On the classification of entire solutions to the critical p -Laplace equation,” 2022, arXiv:2210.05141) under the condition <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>p</m:mi> <m:mo>≥</m:mo> <m:mfrac> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:mfrac> </m:math> p≥ n+1/3 .