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On the existence of extremal functions in Sobolev embedding theorems with critical exponents. (English. Russian original) Zbl 1113.49010

St. Petersbg. Math. J. 17, No. 5, 773-796 (2006); translation from Algebra Anal. 17, No. 5, 105-140 (2005).
Summary: Sufficient conditions for the existence of extremal functions in Sobolev-type inequalities on manifolds with or without boundary are established. Some of these conditions are shown to be sharp. Similar results for embeddings in some weighted \(L_q\)-spaces are obtained.

MSC:

49J40 Variational inequalities
49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
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[1] Adimurthi and G. Mancini, The Neumann problem for elliptic equations with critical nonlinearity, Nonlinear analysis, Sc. Norm. Super. di Pisa Quaderni, Scuola Norm. Sup., Pisa, 1991, pp. 9 – 25. · Zbl 0836.35048
[2] Angelo Alvino, Vincenzo Ferone, Guido Trombetti, and Pierre-Louis Lions, Convex symmetrization and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), no. 2, 275 – 293 (English, with English and French summaries). · Zbl 0877.35040 · doi:10.1016/S0294-1449(97)80147-3
[3] Thierry Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry 11 (1976), no. 4, 573 – 598 (French). · Zbl 0371.46011
[4] Marino Belloni and Bernd Kawohl, The pseudo-\?-Laplace eigenvalue problem and viscosity solutions as \?\to \infty , ESAIM Control Optim. Calc. Var. 10 (2004), no. 1, 28 – 52. · Zbl 1092.35074 · doi:10.1051/cocv:2003035
[5] M. Belloni, V. Ferone, and B. Kawohl, Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators, Z. Angew. Math. Phys. 54 (2003), no. 5, 771 – 783. Special issue dedicated to Lawrence E. Payne. · Zbl 1099.35509 · doi:10.1007/s00033-003-3209-y
[6] G. A. Bliss, An integral inequality, J. London Math. Soc. 5 (1930), 40-46. · JFM 56.0434.02
[7] Haïm Brezis, Some variational problems with lack of compactness, Nonlinear functional analysis and its applications, Part 1 (Berkeley, Calif., 1983) Proc. Sympos. Pure Math., vol. 45, Amer. Math. Soc., Providence, RI, 1986, pp. 165 – 201.
[8] D. Cordero-Erausquin, B. Nazaret, and C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Adv. Math. 182 (2004), no. 2, 307 – 332. · Zbl 1048.26010 · doi:10.1016/S0001-8708(03)00080-X
[9] Olivier Druet, Optimal Sobolev inequalities of arbitrary order on compact Riemannian manifolds, J. Funct. Anal. 159 (1998), no. 1, 217 – 242. · Zbl 0923.46035 · doi:10.1006/jfan.1998.3264
[10] N. Ghoussoub and X. S. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), no. 6, 767 – 793 (English, with English and French summaries). · Zbl 1232.35064 · doi:10.1016/j.anihpc.2003.07.002
[11] N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc. 352 (2000), no. 12, 5703 – 5743. · Zbl 0956.35056
[12] Emmanuel Hebey and Michel Vaugon, Meilleures constantes dans le théorème d’inclusion de Sobolev, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), no. 1, 57 – 93 (French, with English and French summaries). · Zbl 0849.53035
[13] Toshihiro Ishibashi and Shigeaki Koike, On fully nonlinear PDEs derived from variational problems of \?^{\?} norms, SIAM J. Math. Anal. 33 (2001), no. 3, 545 – 569. · Zbl 1030.35088 · doi:10.1137/S0036141000380000
[14] Линейные и квазилинейные уравнения ѐллиптического типа., Издат. ”Наука”, Мосцощ, 1973 (Руссиан). Сецонд едитион, ревисед. Олга А. Ладыженская анд Нина Н. Урал’цева, Линеар анд чуасилинеар еллиптиц ечуатионс, Транслатед фром тхе Руссиан бы Сцрипта Течница, Инц. Транслатион едитор: Леон Ехренпреис, Ацадемиц Пресс, Нещ Ыорк-Лондон, 1968.
[15] Elliott H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), no. 2, 349 – 374. · Zbl 0527.42011 · doi:10.2307/2007032
[16] Chang-Shou Lin, Locating the peaks of solutions via the maximum principle. I. The Neumann problem, Comm. Pure Appl. Math. 54 (2001), no. 9, 1065 – 1095. · Zbl 1035.35039 · doi:10.1002/cpa.1017
[17] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109 – 145 (English, with French summary). P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223 – 283 (English, with French summary). P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109 – 145 (English, with French summary). P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223 – 283 (English, with French summary). · Zbl 0541.49009
[18] P.-L. Lions, F. Pacella, and M. Tricarico, Best constants in Sobolev inequalities for functions vanishing on some part of the boundary and related questions, Indiana Univ. Math. J. 37 (1988), no. 2, 301 – 324. · Zbl 0631.46033 · doi:10.1512/iumj.1988.37.37015
[19] A. I. Nazarov and A. P. Shcheglova, On some properties of extremals in a variational problem generated by the Sobolev embedding theorem, Nonlinear Problems and Function Theory, Probl. Mat. Anal., vyp. 27, Novosibirsk, 2004, pp. 109-136; English transl., J. Math. Sci. 120 (2004), no. 2, 1125-1144. · Zbl 1060.46024
[20] Giorgio Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353 – 372. · Zbl 0353.46018 · doi:10.1007/BF02418013
[21] Neil S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721 – 747. · Zbl 0153.42703 · doi:10.1002/cpa.3160200406
[22] Xu Jia Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations 93 (1991), no. 2, 283 – 310. · Zbl 0766.35017 · doi:10.1016/0022-0396(91)90014-Z
[23] Meijun Zhu, On the extremal functions of Sobolev-Poincaré inequality, Pacific J. Math. 214 (2004), no. 1, 185 – 199. · Zbl 1113.58009 · doi:10.2140/pjm.2004.214.185
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