On the effect of equivalent constraints on a maximizing problem associated with the Sobolev type embeddings in 详细信息    查看全文

• 作者：Michinori Ishiwata ; Hidemitsu Wadade
• 刊名：Mathematische Annalen
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：364
• 期：3-4
• 页码：1043-1068
• 全文大小：570 KB
• 参考文献：1.Adachi, S., Tanaka, K.: A scale-invariant form of Trudinger–Moser inequality and its best exponent. Proc. Am. Math. Soc. 1102, 148–153 (1999)MathSciNet MATH
  2.Cao, D.M.: Nontrivial solution of semilinear elliptic equation with critical exponent in \({\mathbb{R}}^{2}\) . Commun. Partial Differ. Equ. 17, 407–435 (1992)MathSciNet CrossRef MATH
  3.Carleson, L., Chang, S.-Y.A.: On the existence of an extremal function for an inequality of J. Moser. Bull. Sci. Math. 2(110), 113–127 (1986)MathSciNet MATH
  4.Flucher, M.: Extremal functions for the Trudinger–Moser inequality in \(2\) dimensions. Comment. Math. Helv. 67, 471–479 (1992)MathSciNet CrossRef MATH
  5.Ishiwata, M.: Existence and nonexistence of maximizers for variational problems associated with Trudinger–Moser type inequalities in \({\mathbb{R}}^{N}\) . Math. Ann. 351, 781–804 (2011)MathSciNet CrossRef MATH
  6.Kozono, H., Sato, T., Wadade, H.: Upper bound of the best constant of a Trudinger–Moser inequality and its application to a Gagliardo–Nirenberg inequality. Indiana Univ. Math. J. 55, 1951–1974 (2006)MathSciNet CrossRef MATH
  7.Li, Y., Ruf, B.: A sharp Trudinger–Moser type inequality for unbounded domains in \({\mathbb{R}}^{n}\) . Indiana Univ. Math. J. 57, 451–480 (2008)MathSciNet CrossRef MATH
  8.Lin, K.C.: Extremal functions for Moser’s inequality. Trans. Am. Math. Soc. 348, 2663–2671 (1996)CrossRef MATH
  9.Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1970)MathSciNet CrossRef MATH
  10.Nagayasu, S., Wadade, H.: Characterization of the critical Sobolev space on the optimal singularity at the origin. J. Funct. Anal. 258, 3725–3757 (2010)MathSciNet CrossRef MATH
  11.Ogawa, T.: A proof of Trudinger’s inequality and its application to nonlinear Schrodinger equation. Nonlinear Anal. 14, 765–769 (1990)MathSciNet CrossRef MATH
  12.Ogawa, T., Ozawa, T.: Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrodinger mixed problem. J. Math. Anal. Appl. 155, 531–540 (1991)MathSciNet CrossRef MATH
  13.Ozawa, T.: Characterization of Trudinger’s inequality. J. Inequal. Appl. 1, 369–374 (1997)MathSciNet MATH
  14.Ozawa, T.: On critical cases of Sobolev’s inequalities. J. Funct. Anal. 127, 259–269 (1995)MathSciNet CrossRef MATH
  15.Pohozaev, S.I.: The Sobolev embedding in the case \(pl=n\) . In: Proceedings of the Technical Scientific Conference on Advances of Scientific Research (1964/1965). Mathematics Section, Moskov. Energetics Institute, Moscow, pp. 158–170 (1965)
  16.Ruf, B.: A sharp Trudinger–Moser type inequality for unbounded domains in \({\mathbb{R}}^{2}\) . J. Funct. Anal. 219, 340–367 (2005)MathSciNet CrossRef MATH
  17.Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)MathSciNet MATH
  18.Weinstein, M.I.: Nonlinear Schrodinger equations and sharp interpolation estimates. Commun. Math. Phys. 87, 567–576 (1982/1983)
  19.Yudovich, V.I.: Some estimates connected with integral operators and with solutions of elliptic equations. Dok. Akad. Nauk SSSR 138, 804–808 (1961)MathSciNet MATH
  
• 作者单位：Michinori Ishiwata (1)
  Hidemitsu Wadade (2)
  
  1. Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka, 560-8531, Japan
  2. Faculty of Mechanical Engineering, Institute of Science and Engineering, Kanazawa University, Kakuma, Kanazawa, Ishikawa, 920-1192, Japan
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1807

文摘

In this paper, we consider the attainability of a maximizing problem $$\begin{aligned} D:=\sup _{\Vert u\Vert _{H^{1,N}_\gamma }=1}\left( \Vert u\Vert _N^N+\alpha \Vert u\Vert _p^p\right) , \end{aligned}$$where \(N\ge 2\), \(N<p<\infty \), \(\alpha >0\), \(0<\gamma \le N\) and \(\Vert u\Vert _{H^{1,N}_\gamma }=\left( \Vert u\Vert _N^\gamma +\Vert \nabla u\Vert _N^\gamma \right) ^{\frac{1}{\gamma }}\). The existence of a maximizer for D is closely related to the exponent \(\gamma \). In fact, we show that the value $$\begin{aligned} \alpha =\alpha _*:=\inf _{\Vert u\Vert _{H^{1,N}_\gamma }=1}\left( \frac{1-\Vert u\Vert _N^N}{\Vert u\Vert _p^p}\right) \end{aligned}$$is a threshold in terms of the attainability of D. Mathematics Subject Classification 47J30 46E35 26D10 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (19) References1.Adachi, S., Tanaka, K.: A scale-invariant form of Trudinger–Moser inequality and its best exponent. Proc. Am. Math. Soc. 1102, 148–153 (1999)MathSciNetMATH2.Cao, D.M.: Nontrivial solution of semilinear elliptic equation with critical exponent in \({\mathbb{R}}^{2}\). Commun. Partial Differ. Equ. 17, 407–435 (1992)MathSciNetCrossRefMATH3.Carleson, L., Chang, S.-Y.A.: On the existence of an extremal function for an inequality of J. Moser. Bull. Sci. Math. 2(110), 113–127 (1986)MathSciNetMATH4.Flucher, M.: Extremal functions for the Trudinger–Moser inequality in \(2\) dimensions. Comment. Math. Helv. 67, 471–479 (1992)MathSciNetCrossRefMATH5.Ishiwata, M.: Existence and nonexistence of maximizers for variational problems associated with Trudinger–Moser type inequalities in \({\mathbb{R}}^{N}\). Math. Ann. 351, 781–804 (2011)MathSciNetCrossRefMATH6.Kozono, H., Sato, T., Wadade, H.: Upper bound of the best constant of a Trudinger–Moser inequality and its application to a Gagliardo–Nirenberg inequality. Indiana Univ. Math. J. 55, 1951–1974 (2006)MathSciNetCrossRefMATH7.Li, Y., Ruf, B.: A sharp Trudinger–Moser type inequality for unbounded domains in \({\mathbb{R}}^{n}\). Indiana Univ. Math. J. 57, 451–480 (2008)MathSciNetCrossRefMATH8.Lin, K.C.: Extremal functions for Moser’s inequality. Trans. Am. Math. Soc. 348, 2663–2671 (1996)CrossRefMATH9.Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1970)MathSciNetCrossRefMATH10.Nagayasu, S., Wadade, H.: Characterization of the critical Sobolev space on the optimal singularity at the origin. J. Funct. Anal. 258, 3725–3757 (2010)MathSciNetCrossRefMATH11.Ogawa, T.: A proof of Trudinger’s inequality and its application to nonlinear Schrodinger equation. Nonlinear Anal. 14, 765–769 (1990)MathSciNetCrossRefMATH12.Ogawa, T., Ozawa, T.: Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrodinger mixed problem. J. Math. Anal. Appl. 155, 531–540 (1991)MathSciNetCrossRefMATH13.Ozawa, T.: Characterization of Trudinger’s inequality. J. Inequal. Appl. 1, 369–374 (1997)MathSciNetMATH14.Ozawa, T.: On critical cases of Sobolev’s inequalities. J. Funct. Anal. 127, 259–269 (1995)MathSciNetCrossRefMATH15.Pohozaev, S.I.: The Sobolev embedding in the case \(pl=n\). In: Proceedings of the Technical Scientific Conference on Advances of Scientific Research (1964/1965). Mathematics Section, Moskov. Energetics Institute, Moscow, pp. 158–170 (1965)16.Ruf, B.: A sharp Trudinger–Moser type inequality for unbounded domains in \({\mathbb{R}}^{2}\). J. Funct. Anal. 219, 340–367 (2005)MathSciNetCrossRefMATH17.Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)MathSciNetMATH18.Weinstein, M.I.: Nonlinear Schrodinger equations and sharp interpolation estimates. Commun. Math. Phys. 87, 567–576 (1982/1983)19.Yudovich, V.I.: Some estimates connected with integral operators and with solutions of elliptic equations. Dok. Akad. Nauk SSSR 138, 804–808 (1961)MathSciNetMATH About this Article Title On the effect of equivalent constraints on a maximizing problem associated with the Sobolev type embeddings in \({\mathbb {R}}^N\) Journal Mathematische Annalen Volume 364, Issue 3-4 , pp 1043-1068 Cover Date2016-04 DOI 10.1007/s00208-015-1243-7 Print ISSN 0025-5831 Online ISSN 1432-1807 Publisher Springer Berlin Heidelberg Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Mathematics, general Keywords 47J30 46E35 26D10 Industry Sectors Finance, Business & Banking Authors Michinori Ishiwata (1) Hidemitsu Wadade (2) Author Affiliations 1. Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka, 560-8531, Japan 2. Faculty of Mechanical Engineering, Institute of Science and Engineering, Kanazawa University, Kakuma, Kanazawa, Ishikawa, 920-1192, Japan Continue reading... 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