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The Choquard equation and related questions. (English) Zbl 0453.47042


MSC:

47J05 Equations involving nonlinear operators (general)
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
Full Text: DOI

References:

[1] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Analysis, 14, 349-381 (1973) · Zbl 0273.49063
[2] Aubin, T., Problèmes isopérimétriques et espaces de Sobolev, Comptes Rendus Paris, 280, 279-281 (1975) · Zbl 0295.53024
[3] Berestycki, H.; Lions, P. L., Existence d’ondes solitaires dans des problèmes non linéaires du type Klein-Gordon, Comptes Rendus Paris, 287, 503-506 (1978) · Zbl 0391.35055
[4] Berestycki, H.; Lions, P. P., Existence d’ondes solitaires dans des problèmes non linéaires du type Klein-Gordon: 2 ème partie, Comptes Rendus Paris, 288, 395-398 (1979) · Zbl 0397.35024
[5] Berestycki, H.; Lions, P. L., Existence of a ground state in nonlinear equations of the type Klein-Gordon, (Cottle; Gianessi, Variational Inequalities (1979), J. Wiley: J. Wiley New York), To appear · Zbl 0707.35143
[8] Berger, M. S., On the existence and structure of stationary states for a nonlinear Klein-Gordon equation, J. funct. Analysis, 9, 249-261 (1972) · Zbl 0224.35061
[10] Coffman, C. V., Uniqueness of the ground state for Δu − \(u + u^3 = 0\) and a variational characterization of the other solutions, Archs. ration. Mech. Analysis, 46, 81-95 (1972) · Zbl 0249.35029
[11] Fukuda, I.; Tsutsumi, M., On the Yukawa-coupled Klein-Gordon Schrödinger equations in three space dimensions, Proc. Japan Acad., 51, 402-405 (1975) · Zbl 0313.35065
[12] Fukuda, I.; Tsutsumi, M., On coupled Klein-Gordon Schrödinger equations, J. math. Analysis Applic., 66, 358-378 (1978), ii · Zbl 0396.35082
[13] Krasnoselski, M. A., Topological Methods in the Theory of Nonlinear Integral Equations (1964), Macmillan: Macmillan New York · Zbl 0111.30303
[14] Lieb, E. H., Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Studies appl. Math., 57, 93-105 (1977) · Zbl 0369.35022
[15] Lieb, E. H.; Simon, B., The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53, 185-194 (1977)
[18] Nehari, Z., On a nonlinear differential equation arising in nuclear physics, Proc. R. Irish Acad., 62, 117-135 (1963) · Zbl 0124.30204
[19] Rabinowitz, P. H., Variational methods for nonlinear eigenvalue problems, (Eigenvalues of Nonlinear Problems (1974), CIME, Edizioni Gremonese: CIME, Edizioni Gremonese Rome) · Zbl 0212.16504
[20] Reeken, M., General theorem on bifurcation and its application to the Hartree equation of the helium atom, J. Math. Phys. vol. 11, 8, 2505-2512 (1970)
[21] Rosen, G., Minimum value for \(c\) in the Sobolev inequality‖\(ϕ\)‖\(_3\) ⩽ \(c\)‖∇\(ϕ\)‖\(_2\), SIAM J. appl. Math., 21, 30-32 (1971) · Zbl 0201.38704
[22] Ryder, G. H., Boundary value problems for a class of nonlinear differenticial equations, Pacif. J. Math., 22, 447-503 (1967) · Zbl 0152.28303
[23] Strauss, W., Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55, 149-162 (1977) · Zbl 0356.35028
[24] Staurt, C. A., Existence theory for the Hartree equation, Archs. ration. Mech. Analysis, 51, 60-69 (1973) · Zbl 0287.34032
[26] Wiegel, F. W., The interacting Bose fluid: path integral representations and renormalization group approach, (Papadopoulos, G. J.; Denease, J. T., Path Integrals (1978), ASI, Plenum Press: ASI, Plenum Press New York) · Zbl 0435.76062
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