References
Agmon, S., Douglis, A., & L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I. Comm. Pure Applied Math. 12, 623–727 (1959).
Amann, H., Nonlinear operators in ordered Banach spaces and some applications to nonlinear boundary value problems. In Nonlinear Operators and the Calculus of Variations, Bruxelles (1975). Lecture Notes in Math. n∘ 543. Berlin Heidelberg New York: Springer 1976.
Amann, H., Ljusternik-Schnirelman theory and nonlinear eigenvalue problems. Math. Ann. 199, 55–72 (1972).
Ambrosetti, A., On the existence of multiple solutions for a class of nonlinear boundary value problems. Rend. Sem. Univ. Padova 49, 195–204 (1973).
Ambrosetti, A., & P. H. Rabinowitz, Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973).
Anderson, D., & G. Derrick, Stability of time dependent particle like solutions in nonlinear field theories. J. Math. Phys. 11, 1336–1346 (1970); and 12, 945–952 (1971).
Aronson, D. G., & H. F. Weinberger, Nonlinear diffusion in population genetics. Lecture Notes in Math. n∘ 446. Berlin Heidelberg New York: Springer 1975.
Berestycki, H., & P. L. Lions, Existence d'ondes solitaires dans des problèmes non linéaires du type Klein-Gordon. C. R. A. S. Paris, série A, 287, 503–506 (1978); 288, 395–398 (1979).
Berestycki, H., & P. L. Lions, Une méthode locale pour l'existence de solutions positives de problèmes semi-linéaires elliptiques dans RN. J. Anal. Math. 3b8 (1980) 144–187.
Berestycki, H., & P. L. Lions, Nonlinear eigenvalue problems, Lectures Note in Math. N∘ 782, C. Bardos, J. M. Lasry & M. Schatzman editors. Berlin Heidelberg New York: Springer 1980.
Berestycki, H., & P. L. Lions, Existence of a ground state in nonlinear equations of the type Klein-Gordon. Variational Inequalities, Cottle, Gianessi & Lions editors. New York: J. Wiley 1979.
Berestycki, H., & P. L. Lions, Existence of stationary states of nonlinear scalar field equations. Bifurcation phenomena in mathematical physics and related topics. 1980, Bardos, C., & Bessis D., editors. New York: Reidel, 269–292.
Berestycki, H., & P. L. Lions, In preparation.
Berestycki, H., & P. L. Lions, Existence of infinitely many solutions in the “zero mass” case for nonlinear scalar field equations. In preparation.
Berestycki, H., Lions, P. L., & L. A. Peletier, An O.D.E. approach to the existence of positive solutions for semilinear problems in R N . Indiana University Math. J. 30, 141–157 (1981).
Berger, M. S., Nonlinearity and nonlinear functional analysis, Academic Press, New York (1979).
Berger, M. S., On the existence and structure of stationary states for a nonlinear Klein-Gordon equation. J. Funct. Anal. 9, 249–261 (1972).
Bourbaki, N., Eléments de mathématiques livre VI, Intégration. Actual Scient. Ind. Paris: Hermann 1963–1967.
Brezis, H., & T. Kato, Remarks on the Schrödinger operator with singular complex potentials. J. Math. Pures Appl. 58, 137–151 (1979).
Brezis, H., & R. E. L. Turner, On a class of superlinear elliptic problems. Comm. P.D.E. 2, (6) 601–614 (1977).
Brezis, H., & L. Veron, Removable singularities of some nonlinear elliptic equations. To appear.
Browder, F. E., Nonlinear eigenvalue problems and group invariance. In Functional Analysis and Related Fields, Browder, F. E. editor. Springer Verlag (1970), New York. 1–58.
Browder, F. E., Existence theorems for nonlinear partial differential equations. Proc. Symp. Pure Math. 16 A.M.S. Providence (1970), 1–60.
Browder, F. E., Infinite dimensional manifolds and nonlinear elliptic eigenvalue problems. Ann. Math. 82, 459–477 (1965).
Clarke, D. C., A variant of the Lusternik-Schnirelman Theory. Ind. Univ. Math. J. 22, 65–74 (1972).
Coffman, C. V., A minimum maximum principle for a class of nonlinear integral equations. J. Analyse Math. 22, 391–419 (1969).
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. 46 (6), 81–95 (1972).
Coleman, S., The fate of the false vacuum. I—Semi-classical Theory. Phys. Rev. D. 15, 2929 (1977).
Coleman, S., Glazer, V., & A. Martin, Action minima among solutions to a class of Euclidean scalar field equations. Comm. Math. Phys. 58 (2), 211–221 (1978).
Dancer, E. N., Boundary value problems for ordinary differential equations in infinite intervals. Proc. London Math. Soc. 30, 76–94 (1975).
Esteban, M. J., Thèse de 3ème cycle, Université Pierre et Marie Curie, Paris (1980–1981) and paper to appear.
Fife, P. C., Asymptotic states for equations of reaction and diffusion. Bull. A. M. S. 84 (5), 693–726 (1978).
Fife, P. C., & L. A. Peletier, Nonlinear diffusion in population genetics. Arch. Rat. Mech. Anal. 64 (2), 93–109 (1977).
Frampton, P. H., Consequences of vacuum instability in quantum field theory. Phys. Rev. D. 15 (10), 2922–2928 (1977).
Gidas, B., Bifurcation phenomena in mathematical physics and related topics. (1980), Bardos, C., & Bessis D., editors. Dordrecht, Holland: Reidel.
Gidas, B., Wei-Ming, Ni, & L. Nirenberg, Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68, 209–243 (1979).
Gidas, B., Wei-Ming, Ni, & L. Nirenberg. To appear.
Hardy, G. H., Littlewood, J. E., & G. Polya, Inequalities. London: Cambridge Univ. Press 1952.
Hempel, J. A., Multiple solutions for a class of nonlinear elliptic boundary value problems. Indiana Univ. Math. J. 20, 983–996 (1971).
Kato, T., Growth properties of solutions of the reduced wave equation with a variable coefficient. Comm. Pure Applied Math. 12, 403–425 (1959).
Keller, C. W., To appear.
Krasnosel'skii, M. A., Topological methods in the theory of nonlinear integral equations. New York: MacMillan 1964.
Lieb, E. H., Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation. Studies in Applied Math. 57, 93–105 (1977).
Lions, J. L., Problèmes aux limites dans les équations aux dérivées partielles. Presses de l'Univ. de Montréal. Montréal (1962).
Lions, P. L., The Choquard equation and related problems. Nonlinear Analysis T. M. A. 4, 1063–1073 (1980).
Ljusternik, L. A., & L. G. Schnirelman, Topological Methods in the calculus of variations. Paris: Hermann 1934.
Nehary, Z., On a nonlinear differential equation arising in nuclear physics. Proc. Roy. Irish Acad. 62, 117–135 (1963).
Nussbaum, R., Positive solutions of nonlinear elliptic boundary value problems. J. Math. Anal. Appl. 51 (2), 461–482 (1975).
Palais, R. S., Ljusternik-Schnirelman theory on Banach manifolds. Topology 5, 115–132 (1966).
Palais, R. S., Critical point theory and the minimax principle. Proc. Symp. Pure Math. 15, A. M. S., Providence, R. I. 185–212 (1970).
Pohožaev, S. I., Eigenfunctions of the equation Δu + λf(u) = 0. Sov. Math. Doklady 5, 1408–1411 (1965).
Rabinowitz, P. H., Variational methods for nonlinear eigenvalue problems. In Eigenvalues of Non-linear problems. C. I. M. E., Ediz. Cremonese, Rome, 1974.
Rabinowitz, P. H., Variational methods for nonlinear elliptic eigenvalue problems. Indiana Univ. Math. J. 23, 729–754 (1974).
Ryder, G. H., Boundary value problems for a class of nonlinear differential equations. Pacific J. Math. 22 (3), 477–503 (1967).
Strauss, W. A., Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55, 149–162 (1977).
Stuart, C. A., Battelle Inst. Math. report n∘ 75, 1973.
Stuart, C. A., Bifurcation for variational problems when the linearization has no eigenvalues. Preprint.
Synge, J. L., On a certain nonlinear differential equation. Proc. Roy. Irish Acad. 62, 17–41 (1961).
Vainberg, M. M., Variational methods for the study of nonlinear operators. San Francisco: Holden Day, 1964.
Weinberger, M. F., Asymptotic behaviour of a model in population genetics. Indiana Univ. Seminar in Applied Math., 1976–1977. Chadam J. editor, Lecture Notes in Math. 648. Berlin Heidelberg New York: Springer 1978.
L. A. Peletier & J. Serrin, Uniqueness of Positive solution of semilinear equations in R n. Arch. Rational Mechanis Anal. 81, 181–197 (1983).
K. McLeod & J. Serrin, Uniqueness of solutions of semilinear equations. Proc. Nat. Acad. Sci. USA, 78, 6592–6595 (1981).
T. Cazenave & P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations. Comm. Math. Phys. 85, 549–561 (1982).
H. Berestycki & T. Cazenave, Instabilité des états stationnaires dans des équations de Schrödinger et de Klein-Gordon non linéaires. C. R. Ac. Sc. Paris, série I, 293, 489–492 (1981).
H. Berestycki & T. Cazenave, Instability of stationary states in nonlinear Schrödinger or Klein-Gordon equations. To appear.
H. Berestycki, T. Gallouet & O. Kavian, On the equation -Δu = g(u) in R 2. To appear.
P. L. Lions, Principe de concentration-compacité en calcul des variations. C. R. Ac. Sc. Paris, série I, 294, 261–264 (1982).
P. L. Lions, To appear.
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Berestycki, H., Lions, P.L. Nonlinear scalar field equations, II existence of infinitely many solutions. Arch. Rational Mech. Anal. 82, 347–375 (1983). https://doi.org/10.1007/BF00250556
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DOI: https://doi.org/10.1007/BF00250556