On the existence of an extremal function in Sobolev embedding theorems with limit exponent
Algebra i analiz, Tome 17 (2005) no. 5, pp. 105-140.

Voir la notice de l'article provenant de la source Math-Net.Ru

@article{AA_2005_17_5_a3,
     author = {A. V. Demyanov and A. I. Nazarov},
     title = {On the existence of an extremal function in {Sobolev} embedding theorems with limit exponent},
     journal = {Algebra i analiz},
     pages = {105--140},
     publisher = {mathdoc},
     volume = {17},
     number = {5},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AA_2005_17_5_a3/}
}
TY  - JOUR
AU  - A. V. Demyanov
AU  - A. I. Nazarov
TI  - On the existence of an extremal function in Sobolev embedding theorems with limit exponent
JO  - Algebra i analiz
PY  - 2005
SP  - 105
EP  - 140
VL  - 17
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AA_2005_17_5_a3/
LA  - ru
ID  - AA_2005_17_5_a3
ER  - 
%0 Journal Article
%A A. V. Demyanov
%A A. I. Nazarov
%T On the existence of an extremal function in Sobolev embedding theorems with limit exponent
%J Algebra i analiz
%D 2005
%P 105-140
%V 17
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AA_2005_17_5_a3/
%G ru
%F AA_2005_17_5_a3
A. V. Demyanov; A. I. Nazarov. On the existence of an extremal function in Sobolev embedding theorems with limit exponent. Algebra i analiz, Tome 17 (2005) no. 5, pp. 105-140. http://geodesic.mathdoc.fr/item/AA_2005_17_5_a3/

[AM] Adimurthi, Mancini G., “The Neumann problem for elliptic equations with critical nonlinearity”, Nonlinear Analysis, Sc. Norm. Super, di Pisa Quaderni, Scuola Norm. Sup., Pisa, 1991, 9–25 | MR

[AFTL] Alvino A., Ferone V., Trombetti G., Lions P.-L., “Convex symmetrization and applications”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 275–293 | DOI | MR | Zbl

[Aub] Aubin T., “Problèmes isopérimetriques et espaces de Sobolev”, J. Differential Geom., 11 (1976), 573–598 | MR | Zbl

[BK] Belloni M., Kawohl B., “The pseudo-$p$-Laplace eigenvalue problem and viscosity solutions as $p\to\infty$”, ESAIM Contrôle Optim. Calc. Var., 10 (2004), 28–52 | DOI | MR | Zbl

[BFK] Belloni M., Ferone V., Kawohl B., “Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators”, Z. Angew. Math. Phys., 54 (2003), 771–783 | DOI | MR | Zbl

[Bl] Bliss G. A., “An integral inequality”, J. London Math. Soc., 5 (1930), 40–46 | DOI | Zbl

[Br] Brezis H., “Some variational problems with lack of compactness”, Nonlinear Functional Analysis and its Applications (Berkeley, CA, 1983), Proc. Sympos. Pure Math., 45, Part 1, Amer. Math. Soc., Providence, RI, 1986, 165–201 | MR

[CNV] Cordero-Erausquin D., Nazaret B., Villani C., “A mass-transportation approach to sharp Sobolev and Gagliardo–Nirenberg inequalities”, Adv. Math., 182 (2004), 307–332 | DOI | MR | Zbl

[Dr] Druet O., “Optimal Sobolev inequalities of arbitrary order on compact Riemannian manifolds”, J. Funct. Anal., 159 (1998), 217–242 | DOI | MR | Zbl

[GhK] Ghoussoub N., Kang X. S., “Hardy–Sobolev critical elliptic equations with boundary singularities”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 767–793 | DOI | MR | Zbl

[GhY] Ghoussoub N., Yuan C., “Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents”, Trans. Amer. Math. Soc., 352 (2000), 5703–5743 | DOI | MR | Zbl

[HV] Hebey E., Vaugon M., “Meilleures constantes dans le theoreme d'inclusion de Sobolev”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 57–93 | MR | Zbl

[IK] Ishibashi T., Koike S., “On fully nonlinear PDEs derived from variational problems of $L^p$ norms”, SIAM J. Math. Anal., 33 (2001), 545–569 | DOI | MR | Zbl

[ЛУ] Ladyzhenskaya O. A., Uraltseva H. H., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, 2-oe izd., Nauka, M., 1973 | MR

[Lb] Lieb E. H., “Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities”, Ann. of Math. (2), 118:2 (1983), 349–374 | DOI | MR | Zbl

[Lin] Lin C. S., “Locating the peaks of solutions via the maximum principle. I. The Neumann problem”, Comm. Pure Appl. Math., 54 (2001), 1065–1095 | DOI | MR | Zbl

[Ls] Lions P.-L., “The concentration-compactness principle in the calculus of variations. The locally compact case. 1, 2”, Ann. Inst. H. Poincaré Anal. Non-Linéaire, 1 (1984), 109–145, 223–283 | MR | Zbl

[LPT] Lions P.-L., Pacella F., Tricarico M., “Best constants in Sobolev inequalities for functions vanishing on some part of the boundary and related questions”, Indiana Univ. Math. J., 37:2 (1988), 301–324 | DOI | MR | Zbl

[НЩ] Nazarov A. I., Scheglova A. P., “O nekotorykh svoistvakh ekstremali v variatsionnoi zadache, porozhdennoi teoremoi vlozheniya Soboleva”, Nelineinye zadachi i teoriya funktsii, vyp. 27, Novosibirsk, 2004, 109–136 | Zbl

[Tal] Talenti G., “Best constant in Sobolev inequality”, Ann. Mat. Pura Appl. (4), 110 (1976), 353–372 | DOI | MR | Zbl

[Tr] Trudinger N. S., “On Harnack type inequalities and their application to quasilinear elliptic equations”, Comm. Pure Appl. Math., 20 (1967), 721–747 | DOI | MR | Zbl

[W] Wang X. J., “Neumann problems of semilinear elliptic equations involving critical Sobolev exponents”, J. Differential Equations, 93:2 (1991), 283–310 | DOI | MR | Zbl

[Zhu] Zhu M., “On the extremal functions of Sobolev–Poincare inequality”, Pacific J. Math., 214:1 (2004), 185–199 | MR | Zbl