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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

49J45Optimal control problems involving semicontinuity and convergence; relaxation
54D45Local compactness, σ-compactness
49S05Variational principles of physics
35J65Nonlinear boundary value problems for linear elliptic equations
49J20Optimal control problems with PDE (existence)
47J05Equations involving nonlinear operators (general)
58E30Variational principles on infinite-dimensional spaces
81Q05Closed and approximate solutions to quantum-mechanical equations