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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

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
54D45 Local compactness, \(\sigma\)-compactness
49S05 Variational principles of physics
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
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