Trudinger-Moser type inequalities for weighted Sobolev spaces involving fractional dimensions

de Oliveira, José; do Ó, João

doi:10.1090/S0002-9939-2014-12019-3

Trudinger-Moser type inequalities for weighted Sobolev spaces involving fractional dimensions
HTML articles powered by AMS MathViewer

by José Francisco de Oliveira and João Marcos do Ó

Proc. Amer. Math. Soc. 142 (2014), 2813-2828

DOI: https://doi.org/10.1090/S0002-9939-2014-12019-3

Published electronically: May 8, 2014

PDF | Request permission

Abstract:

We derive sharp Trudinger-Moser inequalities for weighted Sobolev spaces and prove the existence of extremal functions. The inequalities we obtain here extend for fractional dimensions the classical results in the radial case. The main ingredient used in our arguments reveals a new proof of a result due to J. Moser for which we give an improved version.

References

Shinji Adachi and Kazunaga Tanaka, Trudinger type inequalities in $\mathbf R^N$ and their best exponents, Proc. Amer. Math. Soc. 128 (2000), no. 7, 2051–2057. MR 1646323, DOI 10.1090/S0002-9939-99-05180-1
Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078
George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $\textbf {R}^2$, Comm. Partial Differential Equations 17 (1992), no. 3-4, 407–435. MR 1163431, DOI 10.1080/03605309208820848
Lennart Carleson and Sun-Yung A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math. (2) 110 (1986), no. 2, 113–127 (English, with French summary). MR 878016
Philippe Clément, Djairo Guedes de Figueiredo, and Enzo Mitidieri, Quasilinear elliptic equations with critical exponents, Topol. Methods Nonlinear Anal. 7 (1996), no. 1, 133–170. MR 1422009, DOI 10.12775/TMNA.1996.006
D. G. de Figueiredo, J. V. Gonçalves, and O. H. Miyagaki, On a class of quasilinear elliptic problems involving critical exponents, Commun. Contemp. Math. 2 (2000), no. 1, 47–59. MR 1753138, DOI 10.1142/S0219199700000049
Djairo G. de Figueiredo, João Marcos do Ó, and Bernhard Ruf, On an inequality by N. Trudinger and J. Moser and related elliptic equations, Comm. Pure Appl. Math. 55 (2002), no. 2, 135–152. MR 1865413, DOI 10.1002/cpa.10015
Djairo G. de Figueiredo, João Marcos do Ó, and Bernhard Ruf, Elliptic equations and systems with critical Trudinger-Moser nonlinearities, Discrete Contin. Dyn. Syst. 30 (2011), no. 2, 455–476. MR 2772124, DOI 10.3934/dcds.2011.30.455
João Marcos B. do Ó, $N$-Laplacian equations in $\mathbf R^N$ with critical growth, Abstr. Appl. Anal. 2 (1997), no. 3-4, 301–315. MR 1704875, DOI 10.1155/S1085337597000419
Martin Flucher, Extremal functions for the Trudinger-Moser inequality in $2$ dimensions, Comment. Math. Helv. 67 (1992), no. 3, 471–497. MR 1171306, DOI 10.1007/BF02566514
Michinori Ishiwata, Makoto Nakamura, and Hidemitsu Wadade, On the sharp constant for the weighted Trudinger-Moser type inequality of the scaling invariant form, Ann. Inst. H. Poincaré C Anal. Non Linéaire 31 (2014), no. 2, 297–314. MR 3181671, DOI 10.1016/j.anihpc.2013.03.004
Jon Jacobsen and Klaus Schmitt, Radial solutions of quasilinear elliptic differential equations, Handbook of differential equations, Elsevier/North-Holland, Amsterdam, 2004, pp. 359–435. MR 2166492
Jon Jacobsen and Klaus Schmitt, The Liouville-Bratu-Gelfand problem for radial operators, J. Differential Equations 184 (2002), no. 1, 283–298. MR 1929156, DOI 10.1006/jdeq.2001.4151
B. Opic and A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics Series, vol. 219, Longman Scientific & Technical, Harlow, 1990. MR 1069756
Yu Xiang Li, Remarks on the extremal functions for the Moser-Trudinger inequality, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 2, 545–550. MR 2214376, DOI 10.1007/s10114-005-0568-7
Kai-Ching Lin, Extremal functions for Moser’s inequality, Trans. Amer. Math. Soc. 348 (1996), no. 7, 2663–2671. MR 1333394, DOI 10.1090/S0002-9947-96-01541-3
Guozhen Lu and Yunyan Yang, Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension, Discrete Contin. Dyn. Syst. 25 (2009), no. 3, 963–979. MR 2533985, DOI 10.3934/dcds.2009.25.963
J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/71), 1077–1092. MR 301504, DOI 10.1512/iumj.1971.20.20101
H. L. Royden, Real analysis, 4th edition, Prentice Hall, New York, 2010.
Michael Struwe, Critical points of embeddings of $H^{1,n}_0$ into Orlicz spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), no. 5, 425–464 (English, with French summary). MR 970849, DOI 10.1016/S0294-1449(16)30338-9
Neil S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483. MR 0216286, DOI 10.1512/iumj.1968.17.17028