Improved fractional Trudinger-Moser inequalities on bounded intervals and the existence of their extremals
Lu Chen, Bohan Wang, Maochun Zhu
Abstract
Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>I</m:mi> </m:math> I be a bounded interval of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="double-struck">R</m:mi> </m:math> {\mathbb{R}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>I</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {\lambda }_{1}\left(I) denote the first eigenvalue of the nonlocal operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mstyle displaystyle="false"> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>4</m:mn> </m:mrow> </m:mfrac> </m:mstyle> </m:mrow> </m:msup> </m:math> {(-\Delta )}^{\tfrac{1}{4}} with the Dirichlet boundary. We prove that for any <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>0</m:mn> <m:mo>⩽</m:mo> <m:mi>α</m:mi> <m:mo><</m:mo> <m:msub> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>I</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> 0\leqslant \alpha \lt {\lambda }_{1}(I) , there holds <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mrow> <m:munder> <m:mrow> <m:mrow> <m:mi>sup</m:mi> </m:mrow> </m:mrow> <m:mrow> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:msubsup> <m:mrow> <m:mi>W</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:mfrac> <m:mo>,</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msubsup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>I</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mo>‖</m:mo> <m:msup> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mstyle displaystyle="false"> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>4</m:mn> </m:mrow> </m:mfrac> </m:mstyle> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:msubsup> <m:mrow> <m:mo>‖</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msubsup> <m:mo>−</m:mo> <m:mi>α</m:mi> <m:msubsup> <m:mrow> <m:mrow> <m:mo stretchy="false">∥</m:mo> <m:mrow>