Points critiques dans les problèmes variationnels sans compacité
Séminaire Bourbaki : volume 1987/88, exposés 686-699, Astérisque, no. 161-162 (1988), Exposé no. 698, 18 p.
@incollection{SB_1987-1988__30__239_0,
     author = {Brezis, Ha{\"\i}m},
     title = {Points critiques dans les probl\`emes variationnels sans compacit\'e},
     booktitle = {S\'eminaire Bourbaki : volume 1987/88, expos\'es 686-699},
     series = {Ast\'erisque},
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     publisher = {Soci\'et\'e math\'ematique de France},
     number = {161-162},
     year = {1988},
     mrnumber = {992212},
     zbl = {0860.58007},
     language = {fr},
     url = {http://archive.numdam.org/item/SB_1987-1988__30__239_0/}
}
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Brezis, Haïm. Points critiques dans les problèmes variationnels sans compacité, dans Séminaire Bourbaki : volume 1987/88, exposés 686-699, Astérisque, no. 161-162 (1988), Exposé no. 698, 18 p. http://archive.numdam.org/item/SB_1987-1988__30__239_0/

[A-R] A. Ambrosetti, P. Rabinowitz - Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), p. 349-381. | MR | Zbl

[A-E] J.-P. Aubin, I. Ekeland - Applied nonlinear analysis, Wiley, New-York (1984). | MR | Zbl

[A 1] Th. Aubin - Problèmes isopérimétriques et espaces de Sobolev, J. Diff. Geom. 11 (1976), p. 573-598. | MR | Zbl

[A 2] Th. Aubin - Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. 55 (1976), p. 269-293. | MR | Zbl

[A 3] Th. Aubin - Nonlinear analysis on manifolds. Monge-Ampère equations, Springer (1982). | MR | Zbl

[Ba 1] A. Bahri - Critical points at infinity in some variational problems, Lecture Notes Longman (à paraître). | Zbl

[Ba-Br] A. Bahri, H. Brezis - Equations elliptiques non linéaires sur des variétés avec exposant de Sobolev critique, C.R. Acad. Sc. Paris (1988) et article détaillé à paraître. | MR | Zbl

[Ba-C 1] A. Bahri, J.-M. Coron - Sur une équation elliptique non linéaire avec l'exposant critique de Sobolev, C.R. Acad. Sc. Paris 301 (1985), p. 345-348). | MR | Zbl

[Ba-C 2] A. Bahri, J.-M. Coron - On a nonlinear elliptic equation involving the critical Sobolev exponent : the effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988), p. 253-294. | MR | Zbl

[Br 1] H. Brezis - Elliptic equations with limiting Sobolev exponents - The impact of the topology, Comm. Pure Appl. Math. 39 (1986), p. 517-539. | MR | Zbl

[Br-C 1] H. Brezis, J.-M. Coron - Convergence of solutions of H-systems or How to blow bubbles, Archive Rat. Mech. Anal. 89 (1985), p. 21-56. | MR | Zbl

[Br-C-N] H. Brezis, J.-M. Coron, L. Nirenberg - Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz, Comm. Pure Appl. Math. 33 (1980), p. 667-689. | MR | Zbl

[Br-N] H. Brezis, L. Nirenberg - Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), p. 437-477. | MR | Zbl

[Co] J.-M. Coron - Topologie et cas limite des injections de Sobolev, C.R. Acad. Sc. Paris 299 (1984), p. 209-212. | MR | Zbl

[Di 1] W. Y. Ding - Positive solutions of Δu + u(n+2)(n-2) = 0 on contractible domains (à paraître).

[Ek] I. Ekeland - On the variational principle, J. Math. Anal. Applic. 47 (1984), p. 324-353. | MR | Zbl

[G-N-N] B. Gidas, W. M. NiL. Nirenberg - Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), p. 209-243. | MR | Zbl

[K-W] J. Kazdan and F. Warner - Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math. 28 (1975), p. 567-597. | MR | Zbl

[Lie] E. Lieb - Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math. 118 (1983), p. 349-374. | MR | Zbl

[Lio] P. L. Lions - The concentration compactness principle in the calculus of variations, the limit case, Rev. Mat. Iberoamericana 1 (1985), p. 145-201 et 1 (1985), p. 45-121. | MR | Zbl

[Mc] B. Mcleod - Communication personnelle.

[N] L. Nirenberg - Variational methods in nonlinear problems, Montecatini CIME Lectures Notes (1987). | Zbl

[O] M. Obata - The conjectures on conformal transformations of Riemannian manifolds, J. Diff. Geom. 6 (1972), p. 247-258. | MR | Zbl

[P-S] R. Palais, S. Smale - A generalized Morse theory, Bull. Amer. Math. Soc. 70 (1984), p. 165-171. | MR | Zbl

[Po] S. Pohozaev - Eigenfunctions of the equation Δu + λ f(u) = 0 , Soviet Mat. Dokl. 6 (1965), p. 1408-1411. | Zbl

[Sa-U] J. Sacks, K. Uhlenbeck - The existence of minimal immersions of 2-spheres, Ann. Math. 113 (1981), p. 1-24. | MR | Zbl

[Sc] R. Schoen - Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geom. 20 (1984), p. 479-495. | MR | Zbl

[Sc-Y 1] R. Schoen, S. T. Yau - On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979), p. 45-76. | MR | Zbl

[Sc-Y 2] R. Schoen, S. T. Yau - Proof of the positive action conjectures in quantum relativity, Phys. Rev. Lett. 42 (1979), p. 547-548.

[St 1] M. Struwe - A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z. 187 (1984), p. 511-517. | MR | Zbl

[Tal] G. Talenti - Best constants in Sobolev inequality, Annali di Mat. 110 (1976), p. 353-372. | MR | Zbl

[Tr] N. Trudinger - Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Sc. Norm. Sup. Pisa 22 (1968), p. 265-274. | Numdam | MR | Zbl

[Y] H. Yamabe - On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), p. 21-37. | MR | Zbl

[J-L] D. Jerison, J. Lee - The Yamabe problem on CR manifold, preprint. | MR

[L-P] J. Lee, T. Parker - The Yamabe problem, Bull. Amer. Math. Soc. 17 (1987), p. 37-81. | MR | Zbl

[Do] S. Donaldson - An application of gauge theory to four-dimensional manifolds, J. Diff. Geom. 18 (1983), p. 279-315. | MR | Zbl

[Se] S. Sedlacek - A direct method for minimizing the Yang-Mills functional over 4-manifolds, Comm. Math. Phys. 86 (1982), p. 515-528. | MR | Zbl

[Tau 1] C. Taubes - The existence of a non-minimal solution to the SU(2) Yang-Mills-Higgs equations on R3 , Comm. Math. Phys. 86 (1982), p. 257-320. | MR | Zbl

[Tau 2] C. Taubes - Path connected Yang-Mills moduli spaces, J. Diff. Geom. 19 (1984), p. 337-392. | MR | Zbl

[Tau 3] C. Taubes - Min-max theory for the Yang-Mills-Higgs equations, preprint. | MR

[Tau 4] C. Taubes - A framework for Morse theory for the Yang-Mills functional, preprint. | MR

[U] K. Uhlenbeck - Variational problems for gauge fields, in Seminar on Differential Geometry (Yau ed.), Princeton Univ. Press (1982), p. 455-464. | MR | Zbl

[Be-C] V. Benci, J.-M. Coron - The Dirichlet problem for harmonic maps from the disk to the Euclidean n-sphere, Ann. IHP, Anallyse non-linéaire 2 (1985), p. 119-141. | Numdam | MR | Zbl

[Br-C 2] H. Brezis, J.-M. Coron - Large solutions for harmonic maps in two dimensions, Comm. Math. Phys. 92 (1983), p. 203-215. | MR | Zbl

[Br-C-L] H. Brezis, J.-M. Coron, E. Lieb - Harmonic maps with defects, Comm. Math. Phys. 107 (1986), p. 649-705. | MR | Zbl

[Cha] K. C. Chang - Heat flow and boundary value problems for harmonic maps, preprint.

[J] J. Jost - The Dirichlet problem for harmonic maps from a surface with boundary onto a 2-sphere, J. Diff. Geom. 19 (1984), p. 393-401. | MR | Zbl

[Sa-U] J. Sacks, K. Uhlenbeck - The existence of minimal immersions of 2-spheres, Ann. Math. 113 (1981), p. 1-24. | MR | Zbl

[St 2] M. Struwe - On the evolution of harmonic mappings, Comm. Math. Helv. 60 (1985), p. 558-581. | MR | Zbl

[St 3] M. Struwe - The evolution of harmonic maps, Part I, II, preprints. | MR

[Ba-R] A. Bahri, P. Rabinowitz - Periodic solutions of the three body problem via the critical points at infinity, preprint.

[Ba 2] A. Bahri - Pseudo-orbits of contactforms, Lecture Notes, Longman (1988). | Zbl

[F] A. Floer - The Conley index for the symplectic action, preprint.

[H] H. Hofer - Symplectic rigidity, holomorphy and Hamiltonian dynamics : a survey, preprint.

[Ba-C 3] A. Bahri, J.-M. Coron - Une théorie des points critiques à l'infini pour l'équation de Yamabe et le problème de Kazdan-Warner, C.R. Acad. Sc. Paris 300 (1985), p. 513-516. | MR | Zbl

[Ba-C 4] A. Bahri, J.-M. Coron - The scalar curvature equation on the standard three-dimensional sphere, preprint

[B-E] J.-P. Bourguignon, J.-P. Ezin - Scalar curvature functions in a conformal class of metrics and conformal transformations, à paraître. | Zbl

[Br-C 2] H. Brezis, J.-M. Coron - Multiple solutions of H-systems and Rellich's conjecture, Comm. Pure Appl. Math. 37 (1984), p. 149-187. | MR | Zbl

[C-Y 1] S. Y. A. Chang, P. C. Yang - Prescribing Gaussian curvature on S 2 , Acta Math. 159 (1987), p. 215-259. | MR | Zbl

[C-Y 2] S. Y. A. Chang, P. C. Yang - Conformal deformation of metrics on S 2 , J. Diff. Geom. 27 (1988), p. 259-296. | MR | Zbl

[C-D] W. Chen, W. Y. Ding - Scalar curvatures on S2, Trans. Amer. Math. Soc. 303 (1987), p. 365-382. | MR | Zbl

[E-S] J. Escobar, R. Schoen - Conformal metrics with prescribed scalar curvature, Invent. Math. 86 (1986), p. 243-254. | MR | Zbl

[Ho] C. W. Hong - A best constant and the Gaussian curvature, Proc. Amer. Math. Soc. 97 (1986), p. 737-747. | MR | Zbl

[K] J. Kazdan - Prescribing the curvature of a Riemannian manifold, CBMS Reg. Conf. Vol. 57, Amer. Math. Soc. (1985). | MR | Zbl

[St 4] M. Struwe - Large H-surfaces via the mountain pass lemma, Math. Ann. 270 (1985), p. 441-459. | MR | Zbl

[St 5] M. Struwe - Non uniqueness in the Plateau problem for surfaces of constant mean curvature, Archive Rat. Mech. Anal. 93 (1986), p. 135-157. | MR | Zbl

[W] H. Wente - Large solutions to the volume constrained Plateau problem, Archive Rat. Mech. Anal. 75 (1980), p. 59-77. | MR | Zbl

[A-Sr] A. Ambrosetti, P. Srikanth - Superlinear elliptic problems and the dual principle in critical point theory, J. Math. and Phys. Sc. 18 (1984), p. 441-451. | MR | Zbl

[A-St] A. Ambrosetti, M. Struwe - A note on the problem - Δu = λu + u |u|2 -2 , Manuscripta Math. 54 (1986), p. 373-379. | Zbl

[At] F. Atkinson - Asymptotics of eigenfunctions for some nonlinear elliptic problems, preprint. | MR

[A-B-P] F. Atkinson, H. Brezis, L. Peletier - Solutions d'équations elliptiques avec exposant de Sobolev critique qui changent de signe, C.R. Acad. Sc. Paris 306 (1988), p. 711-714. | MR | Zbl

[A-P 1] F. Atkinson, L. Peletier - Emden-Fowler equations involving critical exponents, J. Nonlinear Analysis 10 (1986), p. 755-776. | MR | Zbl

[A-P 2] F. Atkinson, L. Peletier - Large solutions of elliptic equations involving critical exponents, Asymptotic Analysis 1 (1988), à paraître. | MR | Zbl

[A-P 3] F. Atkinson, L. Peletier - Elliptic equations with nearly critical growth, J. Diff. Eq. 70 (1987), p. 349-365. | MR | Zbl

[B-P] C. Bandle, L. Peletier - Nonlinear elliptic problems with critical exponents in shrinking annuli, Math. Annalen, à paraître. | Zbl

[Be-C] V. Benci, G. Cerami - Existence of positive solutions of the equation -Δu + a(x)u = u(N+2)/(N-2) in RN , J. Funct. Anal., à paraître. | Zbl

[Br-O] H. Brezis, L. Oswald - A maximization problem involving critical Sobolev exponents, Mat. Aplic. Comp. 6 (1987), p. 47-56. | MR | Zbl

[Br-P] H. Brezis, L. Peletier - Asymptotics for elliptic equations involving critical growth, preprint. | MR

[Bu 1] C. Budd - Semilinear elliptic equations with near critical growth rates, Proc. Roy. Soc. Edinburgh 107 (1987), p. 249-270. | MR | Zbl

[Bu 2] C. Budd - Bifurcation in elliptic systems, preprint. | MR | Zbl

[Bu-N] C. Budd, J. Norbury - Symmetry breaking in semilinear elliptic equations with critical growth rates, preprint.

[C-F-P] A. Capozzi, D. Fortunato, G. Palmieri - An existence result for nonlinear elliptic problems involving critical Sobolev exponents, Ann. IHP, Analyse non-linéaire, 2 (1985), p. 463-470. | Numdam | MR | Zbl

[C-S-S] G. Cerami, S. Solimini, M. Struwe - Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal. 69 (1986), p. 289-306. | MR | Zbl

[Da] E. Dancer - A note on an equation with critical exponent, Bull. London Math. Soc., à paraître. | MR | Zbl

[Di 2] W. Y. Ding - On a conformally invariant elliptic equation on RN , Comm. Math. Phys. 107 (1986), p. 331-335. | MR | Zbl

[Eg 1] H. Egnell - Semilinear elliptic equations involving critical Sobolev exponents, preprint. | MR

[Eg 2] H. Egnell - Elliptic boundary value problems with singular coefficients and critical nonlinearities, preprint. | MR

[Eg 3] H. Egnell - Existence and nonexistence results for m-Laplace equations involving critical Sobolev exponents, preprint. | MR

[Es] J. Escobar - Positive solutions for some semilinear elliptic equations with critical Sobolev exponents, Comm. Pure Appl. Math. 40 (1987), p. 623-657. | MR | Zbl

[F-J] D. Fortunato, E. Janelli - Infinitely many solutions for some nonlinear elliptic problems in symmetrical domains, Proc. Roy. Soc. Edinburgh 105 (1987), p. 205-213. | MR | Zbl

[Jo] C. Jones - Radial solutions of a semilinear elliptic equation at a critical exponent, Archive Rat. Mech. Anal., à paraître. | Zbl

[Le] R. Lewandowski - Little holes and convergence of solutions of -Δu = u(N+2)/(N-2), preprint.

[M-M] G. Mancini, R. Musina - Holes and obstacles, preprint.

[Ni] W. M. Ni - On the elliptic equation -Δu + K(x) u(N+2)/(N-2) = 0 , its generalizations and applications in geometry, Indiana Univ. Math. J. 31 (1982), p. 493-529. | Zbl

[R 1] O. Rey - Le rôle de la fonction de Green dans une équation elliptique non linéaire avec exposant critique de Sobolev, C.R. Acad. Sc. Paris 305 (1987), p. 591-594. | MR | Zbl

The role of Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., à paraître. | Zbl

[R 2] O. Rey - Un résultat de multiplicité dans un problème variationnel non compact, C.R. Acad. Sc. Paris 305 (1987), p. 591-594. | MR | Zbl

A multiplicity result for a variational problem with lack of compactness, J. Nonlinear Anal. | Zbl

[R 3] O. Rey - Sur un problème variationnel non compact : l'effet de petits trous dans le domaine, C.R. Acad. Sc. Paris, à paraître. | Zbl

[So] S. Solimini - On the existence of infinitely many radial solutions for some elliptic problems, Rev. Mat. Apl. (Univ. de Chile) 9 (1987), p. 75-86. | MR | Zbl

[Z] D. Zhang - On multiple solutions of Δu + λu + |u|4/(n-2)u = 0 , preprint.

[A-P 4] F. Atkinson, L. Peletier - Ground states and Dirichlet problems for -Δu = f(u) in R2 , Archive Rat. Mech. Anal. 96 (1986), p. 147-165. | Zbl

[Ca-Ch] L. Carleson, S. Y. A. Chang - On the existence of an external function for an inequality of J. Moser, Bull. Sc. Math., 110 (1986), p. 113-127. | MR | Zbl

[Ch] S. Y. A. Chang - Extremal functions in a sharp form of Sobolev inequality, Proc. Intern. Congress Berkeley (1986), p. 715-723. | MR | Zbl

[Mc-Mc] B. Mcleod, K. Mcleod - The critical Sobolev expònent in two dimensions, preprint. | MR

[Mc-P] B. Mcleod, L. Peletier - Observations on Moser's inequality, preprint.

[St 6] M. Struwe - Critical points of embeddings of H1,n into Orlicz spaces, preprint.

[B-L] H. Brézis, E. Lieb - Sobolev inequalities with remainder terms, J. Funct. Anal. 62 (1985), p. 73-86. | MR | Zbl

[Che] P. Cherrier - Problèmes de Neumann non-linéaires sur les variétés riemanniennes, J. Funct. Anal. 57 (1984), p. 154-206. | MR | Zbl

[Ja] S. Jacobs - An isoperimetric inequality for functions analytic in multiply connected domains, Report Mittag-Leffler Inst. (1970).

[Me-Y] W. Meeks, S. T. Yau - Topology of three-dimensional manifolds and the embedding problems in minimal surface theory, Annals of Math. 112 (1980), p. 441-484. | MR | Zbl

[Si-Y] Y. T. Siu, S. T. Yau - Compact Kähler manifolds of positive bisectional curvature, Invent. Math. 59 (1980), p. 189-204. | MR | Zbl

[Ba 1] A. Barri - Critical points at infinity in some variational problems, Lecture Notes Longman (à paraître). | Zbl

[Ba-C 5] A. Bahri, J.-M. Coron - Vers une théorie des points critiques à l'infini, Sém. Bony-Sjöstrand-Meyer 1984-85. | Numdam | Zbl

[Br 2] H. Brezis - Some variational problems with lack of compactness, Proc. Symp. Pure Math. 45 (1986), Part 1, Amer. Math. Soc., p. 165-201. | MR | Zbl

[Br 3] H. Brezis - Nonlinear elliptic equations involving the critical Sobolev exponent - Survey and perspectives in Directions in Partial Differential Equations (M. Crandall, P. Rabinowitz, R. Turner ed.), Acad. Press (1987), p. 17-36. | MR | Zbl

[Lio] P.L. Lions - The concentration compactness principle in the calculus of variations, the limit case, Rev. Mat. Iberoamericana 1 (1985), p. 145-201 et 1 (1985), p. 45-121. | MR | Zbl