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The concentration-compactness principle in the calculus of variations. The locally compact case. II. (English) Zbl 0704.49004
[For part I see the author, ibid. 109-145 (1984; Zbl 0541.49009) which is also covered by the following review.]
Let H be a function space on \({\mathbb{R}}^ N\), and let J,\({\mathcal E}:\) \(H\to {\mathbb{R}}\), \[ {\mathcal E}(u)=\int e(x,Au(x))dx,\quad J(u)=\int j(x,Bu(x))dx, \] where e: \({\mathbb{R}}^ N\times {\mathbb{R}}^ m\to {\mathbb{R}}\), j: \({\mathbb{R}}^ N\times {\mathbb{R}}^ n\to [0,\infty [\), and A: \(H\to E\), B: \(H\to F\) (E,F are function spaces defined on \({\mathbb{R}}^ N\) with values in \({\mathbb{R}}^ m\), \({\mathbb{R}}^ n\), respectively) commute with a translation of \({\mathbb{R}}^ N\); we consider the minimization problem inf\(\{\) \({\mathcal E}(u):\) \(u\in H\), \(J(u)=1\}\). Because of the loss of boundedness of domains, the classical convexity-compactness methods fail to treat the problem and thus the author presents a new method to solve it. He derives a general principle in a heuristic form and it is rigorously justified on all problems studied in the paper. He first imbeds the problem into a one-parameter family of problems \[ I_{\lambda}=\inf \{{\mathcal E}(u):\;u\in H,\quad J(u)=\lambda \},\quad \lambda >0; \] he supposes \(j(x,q)\to j^{\infty}(q)\), \(e(x,p)\to e^{\infty}(p)\) as \(x\to \infty\) for all \(p\in {\mathbb{R}}^ m\), \(q\in {\mathbb{R}}^ n\); and he considers \[ I^{\infty}_{\lambda}=\inf \{{\mathcal E}^{\infty}(u):\;u\in H,\quad J^{\infty}(u)=\lambda \},\text{ where } {\mathcal E}^{\infty}(u)=\int e^{\infty}(Au(x))dx,\quad J^{\infty}(u)=\int j^{\infty}(B(u(x))dx; \] he assumes \(\{\) \(u\in H:\) \(J(u)=\lambda \}\neq \emptyset\), \(I_{\lambda}>-\infty\) for \(\lambda\in]0,1]\) and that minimizing sequences for \(I_{\lambda}\), \(I^{\infty}_{\lambda}\) are bounded in H.
The concentration-compactness principle is the following: In the case when e and j depend on the first variable, for each \(\lambda >0\) all minimizing sequences for I are relatively compact if and only if the strict subadditivity condition \(I_{\lambda}<I_{\alpha}+I^{\infty}_{\lambda -\alpha}\) holds for all \(\alpha\in [0,\lambda [\); in the case when e and j do not depend on the first variable, for each \(\lambda >0\) all minimizing sequences for I are relatively compact up to a translation if and only if the strict subadditivity condition \(I_{\lambda}<I_{\alpha}+I^{\infty}_{\lambda -\alpha}\) holds for all \(\alpha\in]0,\lambda [\) (he remarks that the weak subadditivity condition is always satisfied). The proof is based upon a compactness lemma obtained with the help of the notion of the concentration function of a measure.
The author gives a rigorous proof of the previous principle in several examples: The rotating star problem: \[ \inf \{\int [j(\rho (x))+k(x)\rho (x)]dx-(1/2)\int \rho (x)\rho (y)f(x-y)dxdy:\;\rho \geq 0,\quad \rho \in L^ 1({\mathbb{R}}^ 3),\quad \int \rho (x)dx=\lambda \} \] where K, f are given, j is a convex function, \(\lambda >0\); the Choquard-Pekar problem: \[ \inf \{\int [(1/2)| \nabla u(x)|^ 2+(1/2)V(x)u(x)^ 2]dx- (1/4)\int u(x)^ 2u(y)^ 2(1/| x-y|)dxdy\}, \] \[ subject\quad to\quad u\in H^ 1({\mathbb{R}}^ N)\text{ and } \int u(x)^ 2dx=1; \] the standing waves in nonlinear Schrödinger equations: \[ \inf \{\int [| \nabla u(x)|^ 2-F(x,u(x))]dx:\;u\in H^ 1({\mathbb{R}}^ N),\quad | u|^ 2_{L^ 2({\mathbb{R}}^ N)}=1\} \] (e.g. \(F(x,t)=| t|^ p)\) and inf\(\{\int [| \nabla u(x)|^ 2+V(x)u(x)^ 2]dx:\) \(u\in H^ 1({\mathbb{R}}^ N)\), \(\int K(x)| u(x)|^ pdx=1\}\), \(p>1\); nonlinear field equations: \[ \inf \{\int | \nabla u(x)|^ 2dx:\;F(x,u(x))dx=\lambda \}; \] unconstrained problems (e.g., Hartree-Fock problems); Euler equations and minimization over manifolds; problems with multiple constraints; problems in unbounded domains other than \({\mathbb{R}}^ N\) (strips, half-spaces, exterior domains, etc.); problems invariant by translation only in some particular directions (e.g., vortex rings, rotating stars).

49J27 Existence theories for problems in abstract spaces
49J10 Existence theories for free problems in two or more independent variables
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Full Text: DOI Numdam EuDML
[1] A. Alvino, P. L. Lions and G. Trombetti, A remark on comparison results for solutions of second order elliptic equations via symmetrization. Preprint. · Zbl 0597.35005
[2] C. J. Amick and J. F. Toland, Nonlinear elliptic eigenvalue problems on an infinite strip: global theory of bifurcation and asymptotic bifurcation. To appear in Math. Ann. · Zbl 0489.35067
[3] Auchmuty, J. F.G., Existence of axisymmetric equilibrium figures, Arch. Rat. Mech. Anal., t. 65, 249-261, (1977) · Zbl 0366.76083
[4] Auchmuty, J. F.G.; Reals, R., Variational solutions of some nonlinear free boundary problems, Arch. Rat. Mech. Anal., t. 43, 255-271, (1971) · Zbl 0225.49013
[5] H. Berestycki, T. Gallouet and O. Kavian, work in preparation.
[6] 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
[7] Berestycki, H.; Lions, P. L., Existence d’ondes solitaires dans des problèmes non linéaires du type klein‐gordon, C. R. Acad. Sci. Paris, t. 288, 395-398, (1979) · Zbl 0397.35024
[8] 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
[9] Berestycki, H.; Lions, P. L., Existence of stationary states in nonlinear scalar field equations, (Bardos, C.; Bessis, D., Bifurcation Phenomena in Mathematical Physics and related topics, (1980), Reidel Dordrecht) · Zbl 0707.35143
[10] Berestycki, H.; Lions, P. L., Existence d’états multiples dans des équations de champs scalaires non linéaires dans le cas de masse nulle, C. R. Acad. Sci. Paris, t. 297, 267-270, (1983) · Zbl 0542.35072
[11] H. Berestycki and P. L. Lions, work in preparation.
[12] Berger, M. S., On the existence and structure of stationary states for a non‐linear Klein-Gordan equation, J. Funct. Anal., t. 9, 249-261, (1972) · Zbl 0224.35061
[13] J. Bona, D. K. Bose and R. E. L. Turner, Finite amplitude steady waves in stratified fluids. To appear in J. Math. Pures Appl. · Zbl 0491.35049
[14] Cazenave, T.; Lions, P. L., Orbital stability of standing waves for some non‐linear Schrödinger equations, Comm. Math. Phys., t. 85, 549-561, (1982) · Zbl 0513.35007
[15] Coffman, C. V., A minimum‐maximum principle for a class of nonlinear integral equations, J. Anal. Math., t. 22, 341-419, (1969) · Zbl 0179.15601
[16] Coffman, C. V., On a class of nonlinear elliptic boundary value problems, J. Math. Mech., t. 19, 351-356, (1970) · Zbl 0194.42103
[17] Coffman, C. V., Uniqueness of the ground state solution for δu - u + u^3 = 0 and a variational characterization of other solutions, Arch. Rat. Mech. Anal., t. 46, 81-95, (1972) · Zbl 0249.35029
[18] 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)
[19] Esteban, M. J., Existence d’une infinité d’ondes solitaires pour des équations de champs non linéaires dans le plan, Ann. Fac. Sc. Toulouse, t. II, 181-191, (1980) · Zbl 0458.35040
[20] Esteban, M. J., Nonlinear elliptic problems in strip‐like domains; symmetry of positive vortex rings, Nonlinear Anal. T. M. A., t. 7, 365-379, (1983) · Zbl 0513.35035
[21] Fraenkel, L. E.; Berger, M. S., A global theory of steady vortex rings in an ideal fluid, Acta Math., 132, 13-51, (1974) · Zbl 0282.76014
[22] Gidas, B.; Ni, W. N.; Nirenberg, L., Symmetry of positive solutions of non‐linear elliptic equations in ℝ^n, (Nachbin, L., Math. Anal. Appl., part 1, (1981), Academic Press)
[23] Lieb, E. H.; Simon, B., The hartree‐fock theory for Coulomb systems, Comm. Math. Phys., t. 53, 185-194, (1974)
[24] Lions, P. L., The concentration‐compactness principle in the calculus of variations. the locally compact case, part 1, Ann. I. H. P. Anal. non linéaire, t. 1, 109-145, (1984) · Zbl 0541.49009
[25] 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)
[26] Lions, P. L., Symmetry and compactness in Sobolev spaces, J. Funct. Anal., t. 49, 315-334, (1982)
[27] Lions, P. L., Principe de concentration‐compacité en calcul des variations, C. R. Acad. Sci. Paris, t. 294, 261-264, (1982) · Zbl 0485.49005
[28] Lions, P. L., On the concentration‐compactness principle, Contributions to Nonlinear Partial Differential Equations, (1983), Pitman London · Zbl 0522.49007
[29] Lions, P. L., Some remarks on Hartree equations, Nonlinear Anal. T. M. A., t. 5, 1245-1256, (1981) · Zbl 0472.35074
[30] Lions, P. L., Minimization problems in L^1(ℝ^N), J. Funct. Anal., t. 49, 315-334, (1982)
[31] P. L. Lions, The concentrationcompactness principle in the Calculus of Variations. The limit case. To appear in Revista Matematica Iberoamericana. · Zbl 0704.49005
[32] Lions, P. L., La méthode de concentration‐compacité en calcul des variations, Séminaire Goulaouic‐Meyer‐Schwartz, 1982‐1983, (1983), École Polytechnique Palaiseau
[33] Lions, P. L., Applications de la méthode de concentration‐compacité à l’existence de fonctions extrêmales, C. R. Acad. Sci. Paris, t. 296, 645-648, (1983) · Zbl 0522.49008
[34] Nehari, Z., On a nonlinear differential equation arising in nuclear physics, Proc. R. Irish Acad., t. 62, 117-135, (1963) · Zbl 0124.30204
[35] Pohozaev, S., Eigenfunctions of the equation δu + λf(u) = 0, Soviet Math. Dokl., t. 165, 1408-1412, (1965) · Zbl 0141.30202
[36] Ryder, G. H., Boundary value problems for a class of nonlinear differential equations, Pac. J. Math., t. 22, 477-503, (1967) · Zbl 0152.28303
[37] Strauss, W., Existence of solitary waves in higher dimensions, Comm. Math. Phys., t. 55, 149-162, (1977) · Zbl 0356.35028
[38] Struwe, M., Multiple solutions of differential equations without the palais‐smale condition, Math. Ann., t. 261, 399-412, (1982) · Zbl 0506.35034
[39] B. R. Suydam, Self‐focusing of very powerful laser beams. U. S. Dept of Commerce. N. B. S. Special Publication 387.
[40] Talenti, G., Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa, t. 3, 697-718, (1976) · Zbl 0341.35031
[41] Taubes, C., The existence of a non‐minimal solution to the SU(2) yang‐mills‐hoggs equations on ℝ^3, Comm. Math. Phys., t. 86, 299, (1982)
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