Sharp Singular Trudinger–Moser Inequalities in Lorentz–Sobolev Spaces

Abstract

Abstract In this paper, we first establish a singular <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo>&lt;</m:mo> <m:mi>β</m:mi> <m:mo>&lt;</m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> ${(0&lt;\beta&lt;n}$ ) Trudinger–Moser inequality on any bounded domain in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> ${\mathbb{R}^{n}}$ with Lorentz–Sobolev norms (Theorem 1.1). Next, we prove the critical singular <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo>&lt;</m:mo> <m:mi>β</m:mi> <m:mo>&lt;</m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> ${(0&lt;\beta&lt;n}$ ) Trudinger–Moser inequality on any unbounded domain in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> ${\mathbb{R}^{n}}$ with Lorentz–Sobolev norms (Theorem 1.2). Then, we set up a subcritical singular <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo>&lt;</m:mo> <m:mi>β</m:mi> <m:mo>&lt;</m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> ${(0&lt;\beta&lt;n}$ ) Trudinger–Moser inequality on any unbounded domain in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> ${\mathbb{R}^{n}}$ with Lorentz–Sobolev norms (Theorem 1.3). Finally, we establish the subcritical nonsingular <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>β</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> ${(\beta=0}$ ) Trudinger–Moser inequality on any unbounded domain in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> ${\mathbb{R}^{n}}$ with Lorentz–Sobolev norms (Theorem 1.5). The constants in all these inequalities are sharp. In [9], for the proof of Theorem 1.2 in the nonsingular case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>β</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> ${\beta=0}$ , the following inequality was used (see [17]): <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:msup> <m:mi>u</m:mi> <m:mo>∗</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>r</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>-</m:mo> <m:mrow> <m:msup> <m:mi>u</m:mi> <m:mo>∗</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>r</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:m