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Zbl 0541.49009
Lions, Pierre-Louis
The concentration-compactness principle in the calculus of variations. The locally compact case. I. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 109-145 (1984). ISSN 0294-1449

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.
[S.Lenhart]
MSC 2000:
*49J45 Optimal control problems inv. semicontinuity and convergence
54D45 Local compactness, etc.
49S05 Variational principles of physics
35J65 (Nonlinear) BVP for (non)linear elliptic equations
49J20 Optimal control problems with PDE (existence)
47J05 Equations involving nonlinear operators (general)
58E30 Variational principles on infinite-dimensional spaces
81Q05 Closed and approximate solutions to quantum-mechanical equations

Keywords: concentration-compactness; minimization problems on unbounded domains; local compactness; rotating stars; Choquard-Pekar problem; nonlinear fields equations

Cited in: Zbl 1247.35141 Zbl 1165.35458 Zbl 1151.35016 Zbl 1208.35033 Zbl 0859.35032 Zbl 0853.35040 Zbl 0838.35035 Zbl 0770.35021 Zbl 0727.35047 Zbl 0759.43001 Zbl 0736.35038 Zbl 0729.35043 Zbl 0702.35095 Zbl 0702.35067 Zbl 0699.35078 Zbl 0707.76026 Zbl 0671.35023 Zbl 0829.49010 Zbl 0696.35053 Zbl 0674.35030 Zbl 0601.49005 Zbl 0704.49004

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