Mean curvature and least energy solutions for the critical Neumann problem with weight
Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 3, pp. 715-733.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

In this paper we consider the Neumann problem involving a critical Sobolev exponent. We investigate a combined effect of the coefficient of the critical Sobolev nonlinearity and the mean curvature on the existence and nonexistence of solutions.
In questo articolo consideriamo il problema di Neumann che richiede un'esponente di Sobolev critico. Noi investighiamo l'effetto combinato del coefficiente della non linearità critica e della curvatura media della frontiera sull'esistenza e sull'inesistenza di soluzioni.
@article{BUMI_2002_8_5B_3_a8,
     author = {Chabrowski, J.},
     title = {Mean curvature and least energy solutions for the critical {Neumann} problem with weight},
     journal = {Bollettino della Unione matematica italiana},
     pages = {715--733},
     publisher = {mathdoc},
     volume = {Ser. 8, 5B},
     number = {3},
     year = {2002},
     zbl = {1097.35046},
     mrnumber = {1205370},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2002_8_5B_3_a8/}
}
TY  - JOUR
AU  - Chabrowski, J.
TI  - Mean curvature and least energy solutions for the critical Neumann problem with weight
JO  - Bollettino della Unione matematica italiana
PY  - 2002
SP  - 715
EP  - 733
VL  - 5B
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BUMI_2002_8_5B_3_a8/
LA  - en
ID  - BUMI_2002_8_5B_3_a8
ER  - 
%0 Journal Article
%A Chabrowski, J.
%T Mean curvature and least energy solutions for the critical Neumann problem with weight
%J Bollettino della Unione matematica italiana
%D 2002
%P 715-733
%V 5B
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BUMI_2002_8_5B_3_a8/
%G en
%F BUMI_2002_8_5B_3_a8
Chabrowski, J. Mean curvature and least energy solutions for the critical Neumann problem with weight. Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 3, pp. 715-733. http://geodesic.mathdoc.fr/item/BUMI_2002_8_5B_3_a8/

[1] Adimurthi-G. Mancini, The Neumann problem for elliptic equations with critical nonlinearity, A tribute in honor of G. Prodi, Scuola Norm. Sup. Pisa (1991), 9-25. | MR | Zbl

[2] Adimurthi-G. Mancini, Effect of geometry and topology of the boundary in critical Neumann problem, J. Reine Angew. Math., 456 (1994), 1-18. | MR | Zbl

[3] Adimurthi-G. Mancini-S. L. Yadava, The role of the mean curvature in a semilinear Neumann problem involving critical exponent, Comm. in P.D.E., 20, No. 3 and 4 (1995), 591-631. | MR | Zbl

[4] Adimurthi-F. Pacella-S. L. Yadava, Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity, J. Funct. Anal., 113 (1993), 318-350. | MR | Zbl

[5] Adimurthi-F. Pacella-S. L. Yadava, Characterization of concentration points and $L^\infty$-estimates for solutions of a semilinear Neumann problem involving the critical Sobolev exponent, Diff. Int. Eq., 8 (1995), 31-68. | Zbl

[6] Adimurthi-S. L. Yadava, Critical Sobolev exponent problem in $\mathbb{R}^{N}$ ($N \geq 4$) with Neumann boundary condition, Proc. Indian Acad. Sci., 100 (1990), 275-284. | MR | Zbl

[7] H. Brézis-L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., 36 (1983), 437-477. | MR | Zbl

[8] J. Chabrowski-M. Willem, Least energy solutions of a critical Neumann problem with weight, to appear in Calc. Var. | Zbl

[9] Z. Djadli, Nonlinear elliptic equations with critical Sobolev exponent on compact riemannian manifolds, Calc. Var., 8 (1999), 293-326. | MR | Zbl

[10] Z. Djadli-O. Druet, Extremal functions for optimal Sobolev inequalities on compact manifolds, Calc. Var., 12 (2001), 59-84. | MR | Zbl

[11] O. Druet, The best constants problem in Sobolev inequalities, Math. Ann., 314 (1999), 327-346. | MR | Zbl

[12] J. F. Escobar, Positive solutions for some nonlinear elliptic equations with critical Sobolev exponents, Commun. Pure Appl. Math., 40 (1987), 623-657. | MR | Zbl

[13] M. Grossi-F. Pacella, Positive solutions of nonlinear elliptic equations with critical Sobolev exponent and mixed boundary conditions, Proc. of the Royal Society of Edinburgh, 116A (1990), 23-43. | MR | Zbl

[14] C. Gui-N. Ghoussoub, Multi-peak solutions for semilinear Neumann problem involving the critical Sobolev exponent, Math. Z., 229 (1998), 443-474. | MR | Zbl

[15] E. Hebey, Sobolev spaces on Riemannian manifolds, Lecture Notes in Mathematics, Springer (1996), 16-35. | MR | Zbl

[16] E. Hebey-M. Vaugon, Meilleures constantes dans le théorème d'inclusion de Sobolev, I.H.P. Analyse non-linéaire, 13 (1996), 57-93. | fulltext mini-dml | MR | Zbl

[17] P. L. Lions, The concentration-compactness principle in the calculus of variations, The limit case, Revista Math. Iberoamericana, 1, No. 1 and No. 2 (1985), 145-201 and 45-120. | MR | Zbl

[18] P. L. Lions-F. Pacella-M. Tricarico, Best constants in Sobolev inequalities for functions vanishing on some part of the boundary and related questions, Indiana Univ. Math. J., 37, No. 2 (1988), 301-324. | MR | Zbl

[19] W. M. Ni-X. B. Pan-L. Takagi, Singular behavior of least energy solutions of a semilinear Neumann problem involving critical Sobolev exponent, Duke Math. J., 67 (1992), 1-20. | fulltext mini-dml | MR | Zbl

[20] W. M. Ni-L. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851. | MR | Zbl

[21] X. J. Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Diff. Eq., 93 (1991), 283-310. | MR | Zbl

[22] Z. Q. Wang, On the shape of solutions for a nonlinear Neumann problem in symmetric domains, Lect. in Appl. Math., 29 (1993), 433-442. | MR | Zbl

[23] Z. Q. Wang, Remarks on a nonlinear Neumann problem with critical exponent, Houston J. Math., 20, No. 4 (1994), 671-694. | MR | Zbl

[24] Z. Q. Wang, High-energy and multipeaked solutions for a nonlinear Neumann problem with critical exponents, Proc. Roy. Soc. of Edinburgh, 125A (1995), 1013-1029. | Zbl

[25] Z. Q. Wang, The effect of the domain geometry on number of positive solutions of Neumann problems with critical exponents, Diff. Int. Eq., 8, No. 6 (1995), 1533-1554. | MR | Zbl

[26] Z. Q. Wang, Construction of multi-peaked solutions for a nonlinear Neumann problem with critical exponent in symmetric domains, Nonl. Anal. T.M.A., 27, No. 11 (1996), 1281-1306. | MR | Zbl

[27] Z. Q. Wang, Existence and nonexistence of $G$-least energy solutions for a nonlinear Neumann problem with critical exponent in symmetric domains, Calc. Var., 8 (1999), 109-122. | MR | Zbl

[28] M. Zhu, Sobolev inequalities with interior norms, Calc. Var., 8 (1999), 27-43. | MR | Zbl