Singular Moser–Trudinger Inequality Involving $L^{n}$ Norm in the Entire Euclidean Space

Abstract

In this paper, we investigate a singular Moser–Trudinger inequality involving $L^{n}$ norm in the entire Euclidean space. The blow-up procedures are used for the maximizing sequence. Then we obtain the existence of extremal functions for this singular geometric inequality in whole space. In general, $W^{1,n}({\mathbb {R}}^n)\hookrightarrow L^q({\mathbb {R}}^n)$ is a continuous embedding but not compact. But in our case we can prove that $W^{1,n}({\mathbb {R}}^n)\hookrightarrow L^n({\mathbb {R}}^n)$ is a compact embedding. Combining the compact embedding $W^{1,n}({\mathbb {R}}^n)\hookrightarrow L^q({\mathbb {R}}^n, |x|^{-s}dx)$ for all $q\ge n$ and $0<s<n$ in [18], we establish the theorems for any $0\le \alpha <1$.

Keywords

Moser–Trudinger inequality / Blow-up analysis / Existence of extremal functions

References

[1] Adimurthi O. Druet, Blow-up analysis in dimension 2 and a sharp form of Trudinger–Moser inequality. Commun. Partial Differ. Equ.. 2004, 29 295-322

[2] Adimurthi K, Sandeep K. A singular Moser–Trudinger embedding and its applications. NoDEA Nolinear Differ. Equ. Appl.. 2007, 13 585-603

[3] Adimurthi Y, Yang HQ. An interpolation of Hardy inequality and Moser-Trudinger in ${\mathbb{R}}^{n}$ and its applications. Int. Math. Res. Not.. 2010, 13 2394-2426

[4] Carleson L, Chang SYA. One the existence of an extremal function for an inequality of J. Moser. Bull. Sci. Math.. 1986, 100 113-127

[5] Figueiredo DEDG, Doò JM, Ruf B. Elliptic equations and systems with critical Trudinger–Moser nonlinearities. Discrete Contin. Dyn. Syst.. 2011, 30 455-476

[6] Flucher M. Extremal functions of for the Trudinger–Moser inequality in two dimensions. Commun. Math. Helv.. 1992, 67 471-497

[7] Heinonen J, Kilpelainen T, Martio O. Nonlinear Potential Theory of Degenerate Elliptic Equations. 1993 Oxford: Oxford University Press

[8] Lam N, Lu GZ. Existence and multiplicity of solutions to equations of n-Laplacian type with critical exponential growth in ${\mathbb{R}}^{n}$. J. Funct. Anal.. 2012, 262 1132-1165

[9] Li YX, Ruf B. A shape Trudinger–Moser type inequality for unbounded domains in ${\mathbb{R}}^{n}$. Indiana Univ. Math. J.. 2008, 57 451-480

[10] Li XM, Yang Y. Extremal functions for singular Trudinger–Moser inequalities in the entire Euclidean space. J. Differ. Equ.. 2018, 264 4901-4943

[11] Lieberman GM. Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. T. M. A.. 1988, 12 1203-1219

[12] Lions PL. The concentration-compactness principle in the calculus of variations. The limit case, part 1. Rev. Math. Iberoamericana. 1985, 1 145-201

[13] Lu GZ, Yang Y. Sharp constant and extremal functiion for the improved Moser–Trudinger inequality involving $L^{p}$ norm in two dimension. Discrete Contin. Dyn. Syst.. 2009, 25 963-979

[14] Lu GZ, Yang Y. A sharpened Moser–Pohozaev–Trudinger inequality with mean value zero in ${\mathbb{R}}^{2}$. Nonlinear Anal.. 2009, 70 2992-3001

[15] Lu, G.Z., Zhu, M.C.: A sharp Trudinger-Moser type inequality involving $L^{n}$ norm in the entire space $\mathbb{R}^{n}$. arXiv:1703.00901v1

[16] Pohozaev, S.: The sobolev embedding in the special case $pl=n$. In: Proceedings of the Technical Scientific Conference on Advances of Scientific Reseach. Mathmatic sections, Moscow Energy Institute, 158-170 (1965)

[17] Ruf B. A sharp Trudinger–Moser inequality for unbounded domains in $R^{2}$. J. Funct. Anal.. 2005, 219 340-367

[18] Serrin J. Local behavior of solutions of quasi-linear equations. Acta. Math.. 1964, 111 247-302

[19] Tolksdorf P. Regularity for a more general class of qusilinear elliptic equations. J. Differ. Equ.. 1984, 51 126-150

[20] Yang YY. A sharp form of Moser–Trudinger inequality in high dimension. J. Funct. Anal.. 2006, 239 100-126

[21] Yuan AF, Zhu XB. An improved singular Trudinger–Moser inequality in unit. J. MAA.. 2016, 435 244-252

[22] Zhang CF, Chen L. Concentration-compactness principle of singular Trudinger–Moser inequalities in $\mathbb{R}^{n}$ and n-Laplace equations. Adv. Nonlinear Stud.. 2018

[23] Zhu JY. Improved Moser–Trudinger inequality involving $L^{p}$ norm in $n$ dimensions. Adv. Nonlinear Stud.. 2014, 14 273-293

Funding

National Natural Science Foundation of China(11771285)

Natural Science Foundation of Jiangxi Province (CN)(GJJ180376)