Sharp Singular Trudinger–Moser Inequalities in Lorentz–Sobolev Spaces

Abstract

AbstractIn this paper, we first establish a singular(0<β<n${(0<\beta<n}$) Trudinger–Moser inequality on any bounded domain inℝn${\mathbb{R}^{n}}$with Lorentz–Sobolev norms (Theorem 1.1). Next, we prove the critical singular(0<β<n${(0<\beta<n}$) Trudinger–Moser inequality on any unbounded domain inℝn${\mathbb{R}^{n}}$with Lorentz–Sobolev norms (Theorem 1.2). Then, we set up a subcritical singular(0<β<n${(0<\beta<n}$) Trudinger–Moser inequality on any unbounded domain inℝn${\mathbb{R}^{n}}$with Lorentz–Sobolev norms (Theorem 1.3). Finally, we establish the subcritical nonsingular(β=0${(\beta=0}$) Trudinger–Moser inequality on any unbounded domain inℝn${\mathbb{R}^{n}}$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 caseβ=0${\beta=0}$, the following inequality was used (see [17]):u∗⁢(r)-u∗⁢(r0)≤1n⁢wn1/n⁢∫rr0U⁢(s)⁢s1/n⁢d⁢ss,$u^{\ast}(r)-u^{\ast}(r_{0})\leq\frac{1}{nw_{n}^{{1/n}}}\int_{r}^{r_{0}}U(s)s^{% {1/n}}\frac{ds}{s},$whereU⁢(x)${U(x)}$is the radial function built from|∇⁡u|${|\nabla u|}$on the level set ofu, i.e.,∫|u|>t|∇u|dx=∫0|{|u|>t}|U(s)ds.$\int_{|u|>t}\lvert\nabla u|\,dx=\int_{0}^{|\{|u|>t\}|}U(s)\,ds.$The construction of suchUuses the deep Fleming–Rishel co-area formula and the isoperimetric inequality and is highly nontrivial. Moreover, this argument will not work in the singular case0<β<n${0<\beta<n}$. One of the main novelties of this paper is that we can avoid the use of this deep construction of such a radial functionU(see remarks at the end of the introduction). Moreover, our approach adapts the symmetrization-free argument developed in [19, 21, 23], where we derive the global inequalities on unbounded domains from the local inequalities on bounded domains using the level sets of the functions under consideration.