Variational Methods pp 161-184 | Cite as

# Lagrange Multipliers, Morses Indices and Compactness

## Abstract

The purpose of this paper is to emphasize two observations on variational problems. In some sense, they are independent even if they can be combined in some problems. The first one concerns the so-called strict subadditivity inequalities in the concentration-compactness method as introduced by the author [23], [24]. See also for some extensions, or applications of these arguments: M.J. Esteban and P.L. Lions [19], [20], M.J. Esteban [17], [18], M. Weinstein [31], P.L. Lions [25], H. Berestycki and P.L. Lions [9], A. Bahri and P.L. Lions [4], P.L. Lions [26], D. Gogny and P.L. Lions [21]. Roughly speaking, this method shows that, for various minimization problems in unbounded domains with constraints, all minimizing sequences are converging if and only if a certain strict subadditivity inequality holds. This inequality involves the infimum of the minimization problem as a function of the “level of the constraint” and corresponds to the possible losses of compactness which are basically due to the effect of “unbounded translations” or “concentrating-diluting dilations.” In some sense, the strict subadditivity inequalities represent the energy balance preventing (and this is necessary and sufficient) losses of compactness. The subadditivity comes into the picture because losses of compactness, when they occur, split the minimizing sequences into various parts which are either “infinitely away from each other” (translations) or “live in different scales” (dilations).

## Keywords

Lagrange Multiplier Morse Index Nonlinear Eigenvalue Problem Grange Multiplier Mors Index## Preview

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## References

- [1]J.F.G. Auchmuty and R. Beals,
*Variational solutions of some nonlinear free boundary problems*, Arch. Rat. Mech. Anal.**43**(1971), 255–271.MathSciNetCrossRefzbMATHGoogle Scholar - [2]A. Bahri,
*Critical points at infinity in some variational problems*,to appear in Longman-Pitman Research Notes.Google Scholar - [3]A. Bahri and J.M. Coron,
*On a nonlinear elliptic equation involving the Sobolev exponent*, preprint.Google Scholar - [4]A. Bahri and P.L. Lions,
*On the existence of a positive solution of semilinear elliptic equations in unbounded domains*, preprint.Google Scholar - [5]A. Bahri and P.L. Lionns,
*Morse indices of some min-max critical points*,*I*,*Applications to multiplicity results*, Comm. Pure Appl. Math.,**61**(1988), 1027–1037.MathSciNetCrossRefGoogle Scholar - [6]A. Bahri and P.L. Lions, in preparation.Google Scholar
- [7]V. Benci and G. Cerami,
*Positive solutions of semilinear elliptic problems in exterior domains*, preprint.Google Scholar - [8]R. Benguria, H. Brézis, and E.H. Lieb,
*The Thomas-Fermi-von Weiziicker theory of atoms and molecules*, Comm. Math. Phys.**79**(1981), 167–180.CrossRefzbMATHGoogle Scholar - [9]H. Berestycki and P.L. Lions,
*Continua of vortex rings*, in preparation.Google Scholar - [10]H. Berestycki and P.L. Lions,
*Nonlinear scalar field equations*, I and II, Arch. Rat. Mech. Anal.**82**(1983), 313–375.MathSciNetzbMATHGoogle Scholar - [11]H. Berestycki and C.H. Taubes, in preparation.Google Scholar
- [12]C.V. Coffman,
*Lyusternik-Schnirelman theory: complementary principles and the Morse index*, Nonlinear Anal. T.M.A.**12**(1988), 507–529.MathSciNetCrossRefzbMATHGoogle Scholar - [13]C.V. Coffman,
*Uniqueness of the ground state solution for*Au–u*+*u3 = 0*and a variational characterization of other solutions*, Arch. Rat. Mech. Anal.**46**(1972), 81–95.Google Scholar - [14]V. Coti-Zelati, I. Ekeland, and P.L. Lions,
*Index estimates and critical points of functionals not satisfying Palais-Smale*, preprint.Google Scholar - [15]I. Ekeland,
*Convexity methods in Hamiltonian systems*, in preparation.Google Scholar - [16]I. Ekeland and H. Hofer, Periodic solutions with prescribed minimal period for convex autonomous Hamiltonian systems, Invent. Math. 81 184 P.L. Lions (1985), 155–188.Google Scholar
- [17]M.J. Esteban,
*A direct variational approach to Skyrme’s model for meson fields*, Comm. Math. Phys.**105**(1986), 187–195.MathSciNetCrossRefGoogle Scholar - [18]M.J. Esteban, this volume.Google Scholar
- [19]M.J. Esteban and P.L. Lions,
*Stationary solutions of nonlinear Schrödinger equations with an external magnetic field*, in*Partial Differential Equations and the Calculus of Variations*, Birkhäuser, Basel, 1989.Google Scholar - [20]M.J. Esteban and P.L. Lions, F
*convergence and the concentration-compactness method for some variational problems with lack of compactness*,preprint.Google Scholar - [21]D. Gogny and P.L. Lions,
*Hartree—Fock theory in Nuclear Physics*, RAIRO Model. Math. et Anal. Num.**20**(1986), 571–637.MathSciNetzbMATHGoogle Scholar - [22]A.C. Lazer and S. Solimini,
*Nontrivial solutions of operator equations and Morse indices of critical points of min-max type*, preprint.Google Scholar - [23]P.L. Lions,
*The concentration-compactness principle in the calculus of variations. The locally compact case*, I and II. Ann. I.H.P. Anal. Non Lin. 1 (1984), 109–145 and 1 (1984), 223–282. See also C.R. Acad. Sci. Paris, 294 (1982), 261–264, and in*Contributions to Nonlinear Partial Differential Equations*, Pitman, London, 1983.Google Scholar - [24]P.L. Lions,
*The concentration-compactness principle in the calculus of variations. The limit case*,Rit. Mat. Iberoamer. 1 (1985), 145–201 and 1 (1985), 45–121. See also C.R. Acad. Sci. Paris, 296 (1983), 645–648 and in*Séminaire Goulaouic-Meyer-Schwartz*,82–83, Ecole Polytechnique, Palaiseau.Google Scholar - [25]P.L. Lions,
*On positive solutions of semilinear elliptic equations in unbounded domains*. In*Nonlinear Diffusion Equations and Their Equilbrium States*, Springer, New York, 1988.Google Scholar - [26]P.L. Lions,
*Solutions of Hartree-Fock equations for Coulomb systems*, Comm. Math. Phys.**109**(1987), 33–97.MathSciNetCrossRefzbMATHGoogle Scholar - [27]P.L. Lions,
*Minimization problems in*L(R), J. Funct. Anal.**41**(1981), 236–275.MathSciNetCrossRefzbMATHGoogle Scholar - [28]S. Solimini,
*Morse index estimates in min-max theories*, preprint.Google Scholar - [29]W. Strauss,
*Existence of solitary waves in higher dimensions*, Comm. Math. Phys.**55**(1977), 149–162.MathSciNetCrossRefzbMATHGoogle Scholar - [30]C. Viterbo,
*Indice de Morse des points critiques obtenus par minimax*, Ann. I.H.P. Anal. Non Lin.**5**(1988), 221–226.MathSciNetzbMATHGoogle Scholar