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The concentration-compactness principle in the calculus of variations. The locally compact case. I. (English) Zbl 0541.49009
This paper presents a general method - called concentration-compactness method - for solving certain minimization problems on unbounded domains. This method applies to problems with some form of local compactness. For minimization problems with constraints, sub-additivity inequalities are obtained for the infimum of the problem considered as a function of the value of the constraint. The concentration-compactness method states that ”all minimizing sequences are relatively compact if and only if the sub- additivity inequalities are strict.” This principle is applied to various examples - rotating stars problem, Choquard-Pekar problem, and nonlinear fields equations.
Reviewer: S.Lenhart

49J45 Methods involving semicontinuity and convergence; relaxation
54D45 Local compactness, \(\sigma\)-compactness
49S05 Variational principles of physics (should also be assigned at least one other classification number in Section 49-XX)
35J65 Nonlinear boundary value problems for linear elliptic equations
49J20 Existence theories for optimal control problems involving partial differential equations
47J05 Equations involving nonlinear operators (general)
58E30 Variational principles in infinite-dimensional spaces
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
Full Text: DOI Numdam EuDML
[1] Auchmuty, J. F.G.; Beals, R., Variational solutions of some nonlinear free boundary problems, Arch. Rat. Mech. Anal., t. 43, 255-271, (1971) · Zbl 0225.49013
[2] Auchmuty, J. F.G.; Beals, R., Models of rotating stars, Astrophys. J., t. 165, 79-82, (1971)
[3] Berestycki, H.; Lions, P. L., Nonlinear scalar field equations. I. existence of a ground state, Arch. Rat. Mech. Anal., t. 82, 313-346, (1983) · Zbl 0533.35029
[4] Berestycki, H.; Lions, P. L., Nonlinear scalar field equations. II. existence of infinitely many solutions, Arch. Rat. Mech. Anal., t. 82, 347-376, (1983) · Zbl 0556.35046
[5] H. Berestycki and P. L. Lions, in preparation.
[6] Berger, M. S., On the existence and structure of stationary states for a nonlinear Klein-Gordon equation, J. Funct. Anal., t. 9, 249-261, (1972) · Zbl 0224.35061
[7] Cazenave, T.; Lions, P. L., Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., t. 85, 549-561, (1982) · Zbl 0513.35007
[8] Coffman, C. V., Uniqueness of the ground state solution for δuu + u^3 = 0 and a variational characterization of other solutions, Arch. Rat. Mech. Anal., t. 46, 81-95, (1972) · Zbl 0249.35029
[9] Coleman, S.; Glazer, V.; Martin, A., Action minima among solutions to a class of Euclidean scalar field equations, Comm. Math. Phys., t. 58, 211-221, (1978)
[10] Donsker and S. R. S. Varadhan, personal communication.
[11] Esteban, M. J.; Lions, P. L., Existence and non-existence results for semilinear elliptic problems in unbounded domains, Proc. Roy. Edim., t. 93, A, 1-14, (1982) · Zbl 0506.35035
[12] Fraenkel, L. E.; Berger, M. S., A global theory of steady vortex rings in an ideal fluid, Acta Math., t. 132, 13-51, (1974) · Zbl 0282.76014
[13] Friedman, A., Variational principles and free-boundary problems, (1982), Wiley New York · Zbl 0564.49002
[14] Lévy, P., Théorie de l’addition des variables aléatoires, (1954), Gauthier-Villars Paris · Zbl 0056.35903
[15] Lieb, E. H., Existence and uniqueness of the minimizing solutions of choquard’s nonlinear equation, Stud. Appl. Math., t. 57, 93-105, (1977) · Zbl 0369.35022
[16] Lions, P. L., Minimization problems in L^1(ℝ^3), J. Funct. Anal., t. 41, 236-275, (1981) · Zbl 0464.49019
[17] Lions, P. L., Compactness and topological methods for some nonlinear variational problems of mathematical physics, (Bishop, A. R.; Campbell, D. K.; Nicolaenko, B., Nonlinear Problems: Present and Future, (1982), North-Holland Amsterdam)
[18] Lions, P. L., Symmetry and compactness in Sobolev spaces, J. Funct. Anal., t. 49, 315-334, (1982)
[19] Lions, P. L., Principe de concentration-compacité en calcul des variations, C. R. Acad. Sci. Paris, t. 294, 261-264, (1982) · Zbl 0485.49005
[20] Lions, P. L., On the concentration-compactness principle, Contributions to Non-linear Partial Differential Equations, (1983), Pitman London · Zbl 0522.49007
[21] Lions, P. L., The Choquard equation and related questions, Nonlinear Anal. T. M. A., t. 4, 1063-1073, (1980) · Zbl 0453.47042
[22] Nehari, Z., On a nonlinear differential equation arising in nuclear physics, Proc. R. Irish Acad., t. 62, 117-135, (1963) · Zbl 0124.30204
[23] Parthasarathy, K. R., Probability measures on metric spaces, (1967), Academic Press New York · Zbl 0153.19101
[24] Ryder, G. H., Boundary value problems for a class of nonlinear differential equations, Pac. J. Math., t. 22, 477-503, (1967) · Zbl 0152.28303
[25] Strauss, W., Existence of solitary waves in higher dimensions, Comm. Math. Phys., t. 55, 149-162, (1977) · Zbl 0356.35028
[26] B. R. Suydam, Self-focusing of very powerful laser beams. U. S. Dept. of Commerce. N. B. S. Special Publication 387.
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