An improved Trudinger–Moser inequality and its extremal functions involving $L^p$ -norm in $mathbb{R}^2$

Let $;W^{1,;2}(mathbb{R}^2)$ be the standard Sobolev space. Denote for any real number p > 2 $lambda p=;begin{array}{c}infuin W^{1,2}(mathbb{R}^2),unotequiv0end{array}$ $frac{int_{mathbb{R}^2}(vertnabla uvert^2+vert uvert^2)dmathcal x}{left(int_{mathbb{R}^2}vert uvert;rho dmathcal xright)^{2/rho}}$Define a norm in $;W^{1,;2}(mathbb{R}^2)$ by${parallel uparallel}_{alpha,p};=;;(int_{mathbb{R}^2}(vertnablaupsilonvert^2+vert uvert^2)dmathcal x-alpha{(int_{mathbb{R}^2}uvert^rho dmathcal x)}^{2/p})^{1/2}$where 0 ≤ α < λp . Using the method of blow-up analysis, we prove that for p > 2 and 0 ≤ α < λp , the supremum$begin{array}{c}supuin W^{1,2;}(mathbb{R}^2),;{parallel uparallel}_{alpha,p}leq1int_{mathbb{R}^{2^{(e^{4pi u2};-;1;-;4pi u^2;)dmathcal x}}}end{array}$can be attained by some function $u_0in W^{1,2}(mathbb{R}^2)withparallel u_0;parallel_{alpha,p}=;1$

PDF

___

[1] Adachi S, Tanaka K. Trudinger type inequalities in R N and their best exponents. Proceedings of the American Mathematical Socitey 2000; 128: 2051-2057. doi: 10.1090/S0002-9939-99-05180-1
[2] Adimurthi, Druet O. Blow-up analysis in dimension 2 and a sharp form of Moser-Trudinger inequality. Communications in Partial Differential Equations 2005; 29 (1-2): 295-322. doi: 10.1081/PDE-120028854
[3] Adimurthi, Sandeep K. A singular Moser-Trudinger embedding and its applications. Nonlinear Differential Equations and Applications 2007; 13 (5): 585-603. doi: 10.1007/s00030-006-4025-9
[4] Adimurthi, Yang Y. An interpolation of Hardy inequality and Trudinger-Moser inequality in R N and its applications. International Mathematics Research Notices 2010; 2010 (13): 2394-2426. doi: 10.1093/imrn/rnp194
[5] Berestycki H, Lions PL. Nonlinear scalar field equations. I. Existence of ground state. Archive for Rational Mechanics and Analysis 1983; 82 (4): 313-345. doi: 10.1007/bf00250555
[6] Cao D. Nontrivial solution of semilinear elliptic equations with critical exponent in R 2 . Communications in Partial Differential Equations 1992; 17 (3): 407-435. doi: 10.1080/03605309208820848
[7] Carleson L, Chang A. On the existence of an extremal function for an inequality of J. Moser. Bulletin des Sciences Mathématiques 1986; 110 (2): 113-127.
8] Chen W, Li C. Classification of solutions of some nonlinear elliptic equations. Duke Mathematical Journal 1991; 63: 615-622.
[9] Csato G, Roy P. Extremal functions for the singular Moser-Trudinger inequality in 2 dimensions. Calculus of Variations and Partial Differential Equations 2015; 54 (2): 2341-2366. doi: 10.1007/s00526-015-0867-5
[10] do Ó JM. N -Laplacian equations in R N with critical growth. Abstract and Applied Analysis 1997; 2 (3-4): 301-315. doi: 10.1155/S1085337597000419
[11] do Ó JM, de Souza M. Trudinger-Moser inequality on the whole plane and extremal functions. Communications in Contemporary Mathematics 2016; 18 (05): 1550054 (2016). doi: 10.1142/S0219199715500546
[12] Hardy G, Littlewood J, Polya G. Inequalities. Cambridge: Cambridge University Press, 1952.
[13] Lu G, Yang Y. Sharp constant and extremal functions for the improved Moser-Trudinger inequality involving L p norms in two dimension. Discrete and Continuous Dynamical Systems 2009; 25 (3): 963-979. doi: 10.3934/dcds.2009.25.963
[14] Flucher M. Extremal functions for Trudinger-Moser inequality in 2 dimensions. Commentarii Mathematici Helvetici 1992; 67 (1): 471-497. doi: 10.1007/bf02566514
[15] Li X. An improved singular Trudinger-Moser inequality in R N and its extremal functions. Journal of Mathematical Analysis and Applications 2018; 462 (2): 1109-1129. doi: 10.1016/j.jmaa.2018.01.080
[16] Li X, Yang Y. Extremal functions for singular Trudinger-Moser inequalities in the entire Euclidean space. Journal of Differential Equations 2018; 264 (8): 4901-4943. doi: 10.1016/j.jde.2017.12.028
[17] Li Y. Moser-Trudinger inequality on compact Riemannian manifolds of dimension two. Journal of partial differential equations 2001; 14 (2): 163-192. doi: 10.1111/j.1536-7150.2004.00338.x
[18] Li Y, Ruf B. A sharp Trudinger-Moser type inequality for unbounded domains in R N . Indiana University Mathematics Journal 2008; 57 (1): 451-480. doi: 10.1512/iumj.2008.57.3137
[19] Lin K. Extremal functions for Moser’s inequality. Transactions of the American Mathematical Society 1996; 348 (7): 2663-2671. doi:10.2307/2155264
[20] Moser J. A sharp form of an inequality by N.Trudinger. Indiana University Mathematics Journal 1971; 20 (11): 1077-1092. doi: 10.1512/iumj.1971.20.20101
[21] Panda R. Nontrivial solution of a quasilinear elliptic equation with critical growth in R n . Proceedings of the Indian Academy of Sciences - Section A 1995; 105 (4): 425-444. doi: 10.1007/BF02836879
[22] Peetre J. Espaces d’interpolation et theoreme de Soboleff. Université de Grenoble. Annales de l’Institut Fourier 1966; 16 (1): 279-317 (in French). doi: 10.5802/aif.232
[23] Pohozaev S. The Sobolev embedding in the special case pl = n. Proceedings of the Technical Scientific Conference on Advances of Scientific Reseach 1964-1965, Mathematics Sections, 158-170, Moscow: Moskov. Energet. Inst., 1965.
[24] Ruf B. A sharp Trudinger-Moser type inequality for unbounded domains in R 2 . Journal of Functional Analysis 2005; 219 (2): 340-367. doi: 10.1016/j.jfa.2004.06.013
[25] Struwe M. Positive solution of critical semilinear elliptic equations on non-contractible planar domain. Journal of the European Mathematical Society 2000; 2 (4): 329-388. doi: 10.1007/s100970000023
[26] Struwe M. Critical points of embeddings of H 1,n 0 into Orlicz spaces. Annales de l Institut Henri Poincare (C) Non Linear Analysis 1988; 5 (5): 425-464. doi: 10.1016/S0294-1449(16)30338-9
[27] Trudinger N. On imbeddings into Orlicz spaces and some applications. Journal of Mathematics and Mechanics 1967; 17 (5): 473-483. doi: 10.1512/iumj.1968.17.17028
[28] Yang Y. A sharp form of Moser-Trudinger inequality in high dimension. Journal of Functional Analysis 2006; 239 (1): 100-126. doi: 10.1016/j.jfa.2006.06.002
[29] Yang Y. Corrigendum to ”A sharp form of Moser-Trudinger inequality in high dimension” [J. Funct. Anal. 239 (2006) 100-126]. Journal of Functional Analysis 2007; 242 (2): 669-671. doi: 10.1016/j.jfa.2006.09.006
[30] Yang Y. A sharp form of the Moser-Trudinger inequality on compact Riemannian surface. Transactions of the American Mathematical Society 2007; 359 (12): 5761-5777. doi: 10.1090/S0002-9947-07-04272-9
[31] Yang Y. Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space. Journal of Functional Analysis 2012; 262 (4): 1679-1704. doi: 10.1016/j.jfa.2011.11.018
[32] Yang Y. Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two. Journal of Differential Equations 2015; 258 (9): 3161-3193. doi: 10.1016/j.jde.2015.01.004
[33] Yang Y, Zhu X. Blow-up analysis concerning singular Trudinger-Moser inequalities in dimension two. Journal of Functional Analysis 2017; 272 (8): 3347-3374. doi: 10.1016/j.jfa.2016.12.028
[34] Yuan A, Zhu X. An improved singular Trudinger-Moser inequality in unit ball. Journal of Mathematical Analysis and Applications 2016; 435 (1): 244-252. doi: 10.1016/j.jmaa.2015.10.038
[35] Yudovich VI. Some estimates connected with integral operators and with solutions of elliptic equations. Doklady Akademii Nauk Sssr 1961; 138 (4): 805-808 (in Russian). doi: 10.1111/j.1749-6632.1961.tb20184.x