Partial differential equations
Solutions to a nonlinear Neumann problem in three-dimensional exterior domains
Comptes Rendus. Mathématique, Volume 356 (2018) no. 9, pp. 933-956.

We prove the existence of multipeak solutions to a nonlinear elliptic Neumann problem involving nearly critical Sobolev exponent, in three-dimensional exterior domains.

Nous démontrons, pour un problème elliptique de Neumann avec non-linéarité presque critique, dans un domaine extérieur de dimension trois, l'existence de solutions qui se concentrent en plusieurs points de la frontière lorsque la non-linéarité devient critique.

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DOI: 10.1016/j.crma.2018.07.005
Olivier, Adélaïde 1; Rey, Olivier 2

1 Laboratoire de mathématiques d'Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France
2 IHPST, Université Paris-1 Panthéon-Sorbonne, CNRS, 13, rue du Four, 75006 Paris, France
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Olivier, Adélaïde; Rey, Olivier. Solutions to a nonlinear Neumann problem in three-dimensional exterior domains. Comptes Rendus. Mathématique, Volume 356 (2018) no. 9, pp. 933-956. doi : 10.1016/j.crma.2018.07.005. http://www.numdam.org/articles/10.1016/j.crma.2018.07.005/

[1] Adimurthi; Pacella, F.P.; Yadava, S.L. Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity, J. Funct. Anal., Volume 113 (1993) no. 2, pp. 318-350

[2] Aubin, T. Problèmes isopérimétriques et espaces de Sobolev, J. Differ. Geom., Volume 11 (1976) no. 4, pp. 573-598

[3] Bahri, A. Critical Points at Infinity in Some Variational Problems, Pitman Res. Notes Math. Ser., vol. 182, Longman House, Harlow, UK, 1989

[4] Bahri, A.; Coron, J.-M. On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Commun. Pure Appl. Math., Volume 41 (1988) no. 3, pp. 253-294

[5] Berger, M.; Gauduchon, P.; Mazet, E. Le spectre d'une variété riemanienne, Lecture Notes in Mathematics, vol. 194, Springer, New York–Berlin, 1971

[6] Brezis, H.; Nirenberg, L. Positive solutions of nonlinear equations involving critical Sobolev exponents, Commun. Pure Appl. Math., Volume 36 (1983) no. 4, pp. 437-477

[7] Caffarelli, L.; Gidas, B.; Spruck, J. Asymptotic symmetry and local behaviour of semilinear elliptic equations with critical Sobolev growth, Commun. Pure Appl. Math., Volume 42 (1989) no. 3, pp. 271-297

[8] Dancer, E.N.; Yan, S. Interior and boundary peak solutions for a mixed boundary value problem, Indiana Univ. Math. J., Volume 48 (1999) no. 4, pp. 1177-1212

[9] Gilbarg, D.; Trudinger, N. Elliptic Partial Differential Equations of Second Order, Grundlehren der mathematischen Wissenschaften, vol. 224, Springer, Berlin, Heidelberg, New York, 1977

[10] Rey, O. The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., Volume 89 (1990) no. 1, pp. 1-52

[11] Rey, O. Boundary effect for an elliptic Neumann problem with critical nonlinearity, Commun. Partial Differ. Equ., Volume 22 (1997) no. 7–8, pp. 1055-1139

[12] Rey, O. An elliptic Neumann problem with critical nonlinearity in three-dimensional domains, Commun. Contemp. Math., Volume 1 (1999) no. 3, pp. 405-449

[13] Talenti, G. Best constants in Sobolev inequality, Ann. Mat. Pura Appl., Volume 110 (1976), pp. 353-372

[14] Wei, J.; Yan, S. Arbitrary many boundary peak solutions for an elliptic Neumann problem with critical growth, J. Math. Pures Appl., Volume 88 (2007) no. 4, pp. 350-378

[15] Yan, S. Multipeak solutions for a nonlinear Neumann problem in exterior domains, Adv. Differ. Equ., Volume 7 (2002) no. 8, pp. 919-950

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