Critical points and nonlinear variational problems
Mémoires de la Société Mathématique de France, no. 49 (1992) , 144 p.
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Ambrosetti, Antonio. Critical points and nonlinear variational problems. Mémoires de la Société Mathématique de France, Série 2, no. 49 (1992), 144 p. doi : 10.24033/msmf.362. http://numdam.org/item/MSMF_1992_2_49__1_0/

[1] Alvino A.-Lions P.L.-Trombetti G., Comparaison des solutions d'équations paraboliques et elliptiques par symétrisation. Une méthode nouvelle, C.R.A.S. 303 (1986), 975-978. | Zbl

[2] Amann H.-Hess P., A multiplicity result for a class of elliptic boundary value problems Proc. Royal. Soc. Ed. 84, A (1979), 145-151. | Zbl | MR

[3] Amann H.-Zehnder E., Nontrivial solutions for a class of non-resonance problems and applications to nonlinear differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4) 7 (1980), 539-603. | Zbl | MR | Numdam | EuDML

[4] Ambrosetti A., Esistenza di infinite soluzioni per problemi non lineari in assenza di parametro, Atti Acc. Naz. Lincei, 52 (1972), 660-667. | Zbl | MR

[5] Ambrosetti A., A perturbation theorem for superlinear boundary value problems, M.R.C. Tech. Summ. Rep. (1974).

[6] Ambrosetti A., Elliptic equations with jumping nonlinearities, J. Math. Phys. Sci. 18-1 (1984), 1-12. | Zbl | MR

[7] Ambrosetti A., Remarks on dynamical systems with singular potentials, in Nonlinear Analysis, a trubute in honour of Giovanni Prodi, Quaderni Scuola Norm. Sup. Pisa, (1991), 51-60. | MR

[8] Ambrosetti A.-Badiale M., The dual variational principle and elliptic problems with discontinuous nonlinearities, Journ. Math. Anal. Appl. 140, 2 (1989), 363-373. | Zbl | MR

[9] Ambrosetti A.-Bessi U., Multiple periodic trajectories in a relativistic gravitational field, in Variational Methods (Ed.H. Berestycki et al.), Birkäuser, (1990), 373-381. | Zbl | MR

[10] Ambrosetti A.-Bessi U., Multiple closed orbits for perturbed Keplerian problems, J. Diff. Equat. to appear. Preliminary note in Rend. Mat. Acc. Lincei, s. 9, v. 2 (1991), 11-15. | Zbl | MR

[11] Ambrosetti A.-Bertotti M.L., Homoclinics for a second order conservative systems, Proc. Conf. in honour of L. Nirenberg, Trento 1990, to appear. | Zbl

[12] Ambrosetti A.-Calahorrano M. - Dobarro F., Remarks on the Grad-Shafranov equation, Appl. Math. Lett. 3-3 (1990), 9-11. | Zbl | MR

[13] Ambrosetti A.-Coti Zelati V., Critical point with lack of compactness and singular dynamical systems, Ann. Mat. Pura Appl. 149 (1987), 237-259. | Zbl | MR

[14] Ambrosetti A.-Coti Zelati V., Perturbation of Hamiltonian Systems with Keplerian Potentials, Mat. Zeit. 201 (1989), 227-242. Preliminary Note in C. R. Acad. Sci. Paris 307 (1988), 568-571. | Zbl | MR

[15] Ambrosetti A.-Coti Zelati V., Closed orbits with fixed energy for singular Hamiltonian Systems, Archive Rat. Mech. Analysis, 112 (1990), 339-362. | Zbl | MR

[16] Ambrosetti A.-Coti Zelati V., Closed orbits with fixed energy for a class of N-body problems, Annales Inst H. Poincaré Analyse Nonlin, to appear. | Zbl | Numdam

[17] Ambrosetti A.-Coti Zelati V. - Ekeland I., Symmetry breaking in Hamiltonian Systems, J. Diff. Equat. 67 (1987), 165-184. | Zbl | MR

[18] Ambrosetti A.-Ekeland I., Periodic solution of a class of Hamiltonian Systems with singularities, Proc. Royal Soc. Edinburgh 114 A (1990), 1-13. | Zbl | MR

[19] Ambrosetti A.-Lupo D., A class of nonlinear Dirichlet with multiple solutions, J. Nonlinear Anal. T.M.A..8-10 (1984), 1145-1150. | Zbl | MR

[20] Ambrosetti A.-Mancini G., Sharp nonuniqueness results for some nonlinear problems, J. Nonlinear Anal. T.M.A. 3 (1979), 635-645. | Zbl | MR

[21] Ambrosetti A.-Mancini G., Remarks on some free boundary problems, Contribution to nonlinear Partial Differential Equations, Pitman (1981), 24-36. | Zbl | MR

[22] Ambrosetti A.-Mancini G., Solution of minimal period for a class of convex Hamiltonian systems, Math. Annalen 255 (1981), 405-421. | Zbl | MR

[23] Ambrosetti A.-Mancini G., On a theorem by Ekeland and Lasry concerning the number of periodic Hamiltonian trajectories, J. Diff. Equat.43 (1982), 249-256. | Zbl | MR

[24] Ambrosetti A.-Prodi G., On the inversion of some differentiable mapping with singularities between Banach spaces, Ann. Mat. Pura Appl. 93 (1972), 231-246. | Zbl | MR

[25] Ambrosetti A.-Prodi G., A primer of Nonlinear Analysis, Cambridge Univ. Press, to appear. | Zbl

[26] Ambrosetti A.-Rabinowitz P.H., Dual variational methods in critical points theory and applications, J. Funct. Anal. 14 (1973), 349-381. | Zbl | MR

[27] Ambrosetti A.-Srikanth P.N., Superlinear elliptic problems and the dual principle in critical point theory, J. Math. Phys. Sci. 18-4 (1984), 441-451. | Zbl | MR

[28] Ambrosetti A.-Struwe M., A note on the problem , Manus Math. 54 (1986), 373-379. | Zbl | MR

[29] Ambrosetti A.-Struwe M., Existence of steady vortex rings in an ideal fluid, Arch. Rat. Mech. Anal. 108, 2 (1989), 97-109. | Zbl | MR

[30] Ambrosetti A.-Turner R.E.L., Some discontinuous variationals problems, Diff. and Integral. Equa. 1 (1988), 341-349. | Zbl | MR

[31] Ambrosetti A.-Yang Janfu, Asymptotic behaviour in planar vortex theory, Rend. Mat. Acc. Lincei, s.9-1 (1990), 285-291. | Zbl

[32] Amick C.J.-Fraenkel L.E., The uniqueness of Hill's spherical vortex, Arch. Rat. Mech. Anal. 2 (1986), 91-119. | Zbl | MR

[33] Amick C.J.-Turner R.E.L., A global branch of steady vortex rings, J. Reine Angw. Math. 384 (1988), 1-23. | Zbl | MR

[34] Bahri A., Topological results on a certain class of functionals and applications, J. Funct. Anal. 41 (1981), 397-427. | Zbl | MR

[35] Bahri A.-Berestycki H., A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc. 267 (1981), 1-32. | Zbl | MR

[36] Bahri A.-Coron J.M., On a nonlinear elliptic equation involving the critical Sobolev exponent : the effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988), 253-294. | Zbl | MR

[37] Bahri A.-Lions P.L., Morse index of some min-max critical points I. Application to multiplicity results, Comm. Pure Appl. Math. 41 (1988), 1027-1037. | Zbl | MR

[38] Bahri A.-Rabinowitz P.H., A minimax method for a class of Hamiltonian systems with singular potentials, J. Funct. Anal. 82 (1989), 412-428. | Zbl | MR

[39] Bahri A.-Rabinowitz P.H., Solutions of the three-body problem via critical points of infinity, Preprint.

[40] Bandle C., Isoperimetric Inequalities and Applications, Pitman, London (1980). | Zbl | MR

[41] Benci V., A geometrical index for the group S1 and some applications to the research of periodic solutions of O.D.E.'s, Comm. Pure Appl. Math. 34 (1981), 393-432. | Zbl | MR

[42] Benci V.-Cerami G., The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Rat. Mech. Anal. 114 (1991), 79-93. | Zbl | MR

[43] Benci V.-Giannoni F., Periodic solutions of prescribed energy for a class of Hamiltonian systems with singular potentials, J. Diff. Eq. 82 (1989), 60-70. | Zbl | MR

[44] Benci V.-Rabinowitz P.H., Critical point theorems for indefinite functionals, Invent. Math. 52 (1979), 241-273. | Zbl | MR

[45] Berestycki H.-Lasry J.M.-Mancini G.-Ruf B., Existence of multiple periodic orbits on star-shaped Hamiltonian surfaces, Comm. Pure Appl. Math. (1985), 253-290. | Zbl | MR

[46] Berger M.S. - Fraenkel L.E., Nonlinear desingularization in certain free-boundary problems, Comm. Math. Phys. 77 (1980), 149-172. | Zbl | MR

[47] Bessi U., Multiple closed orbits for singular conservative systems via geodesics theory, Rend. Sem. Mat. Univ. Padova, to appear. | Zbl | Numdam

[48] Bessi U., Multiple closed orbits of fixed energy for gravitational potentials, J. Diff. Eq., to appear. | Zbl

[49] Bessi U., Multiple homoclinics for autonomous singular potentials, to appear.

[50] Bessi U. - Coti Zelati V., Symmetries and noncollision closed orbits for a planar N-body type problems, J. Nonlin. Anal. T.M.A. 16 (1991), 587-598. | Zbl | MR

[51] Brezis H. - Coron J.M. - Nirenberg L., Free vibrations for a nonlinear equation and a theorem of P. Rabinowitz, Comm. Pure Appl. Math. 33 (1980), 667-684. | Zbl | MR

[52] Brezis H. - Nirenberg L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437-477. | Zbl | MR

[53] Browder F., Infinite dimensional manifolds and nonlinear eigenvalue problems, Ann. of Math. 82 (1965), 459-477. | Zbl | MR

[54] Candela A.M., Remarks on the number of positive solutions for a class of nonlinear elliptic problems, Diff. & Int. Eq. (to appear). | Zbl

[55] Capozzi A. - Fortunato D. - Palmieri G., An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. Poincaré, Anal. Nonlin. 2 (1985), 463-470. | Zbl | MR | Numdam

[56] Capozzi A. - Greco C. - Salvatore A., Lagrangian systems in presence of singularities, Proc. Am. Math. Soc. 102 (1988), 125-130. | Zbl | MR

[57] Clarke F.H., Periodic solutions of Hamiltonian inclusions, J. Diff. Equat. 40 (1981), 1-6. | Zbl | MR

[58] Clarke F.H. - Ekeland I., Hamiltonian trajectories having prescribed minimal period, Comm. Pure Appl. Math. 33 (1980), 103-116. | Zbl | MR

[59] Chang C.K., Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102-129. | Zbl | MR

[60] Chang C.K., A remark on the perturbation of critical manifolds, Preprint Peking University.

[61] Cerami G., Soluzioni positive di problemi con parte nonlienare discontinua e applicazioni a un problema di frontiera libera, Boll. U.M.I. 2 (1983), 321-338. | Zbl | MR

[62] Coron J.M., Topologie et cas limite des injections de Sobolev, C.R. Acad. Sci. Paris 299 (1984), 209-212. | Zbl | MR

[63] Coti Zelati V., Periodic solutions for a class of planar, singular dynamical systems, J. Math. Pures et Appl. 68 (1989), 109-119. | Zbl | MR

[64] Coti Zelati V., A class of periodic solutions of the N-body problem, Cel. Mech. and Dyn. Astr. 46 (1989), 177-186. | Zbl | MR

[65] Coti Zelati V., Periodic solutions for N-body type problems, Annales Inst. H. Poincaré, Analyse Nonlin. 7 (1990), 477-492. | Zbl | MR | Numdam

[66] Coti Zelati V. - Ekeland I. - Séré E., A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann. Vol.288 (1990), 133-160. | Zbl | MR

[67] Coti Zelati V. - Rabinowitz P.H., Homoclinic orbits for a second order Hamiltonian systems possessing superquadratic potentials, Preprint SISSA, Trieste (1980).

[68] Coti Zelati V. - Serra E., Collision and non-collision solutions for a class of Keplerian-like dynamical systems, Preprint SISSA, Trieste (1990).

[69] Dancer E.N., Counterexamples to some conjectures on the number of solutions of nonlinear equations, Math. Ann. 272 (1985), 421-440. | Zbl | MR

[70] Dancer E.N., The G-invariant implicit function theorem in infinite dimension, Proc. Roy. Soc. Edinburgh, 102-A (1986), 211-220. | Zbl | MR

[71] Degiovanni M.-Giannoni F., Periodic solutions of dynamical systems with Newtonian type potentials, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1988), 467-494. | Zbl | MR | Numdam

[72] Ekeland I., Nonconvex minimization problems, Bull. Am. Math. Soc. 1 (1979), 443-474. | Zbl | MR

[73] Ekeland I., Convexity methods in hamiltonian mechanics, Springer, 1990. | Zbl | MR

[74] Ekeland I.-Hofer H., Periodic solution with prescribed minimal period for convex autonomous hamiltonian systems, Invent. Math. 81 (1985), 155-188. | Zbl | MR

[75] Ekeland I.-Lasry J.M., On the number of closed trajectories for a hamiltonian flow on a convex energy surface, Ann. of Math. 112 (1980), 283-319. | Zbl | MR

[76] Fadell E.-Husseini S., A note on the category of free loop space, Proc. A.M.S. 107 (1989), 527-536. | Zbl | MR

[77] Fraenkel L.E.-Berger M.S., A global theory of steady vortex rings in an ideal fluid, Acta Math. 132 (1974), 13-51. | Zbl | MR

[78] Fucik S., Solvability of nonlinear equations and boundary value problems, D. Reidel Publ. Co., Dordrecht (1980). | Zbl | MR

[79] Gallouet T.-Kavian O., Resonance for jumping nonlinearities, Comm. P.D.E. 7-3 (1982), 325-342. | Zbl | MR

[80] Gidas B.-Ni W.M.-Nirenberg L., Symmetry and relates properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243. | Zbl | MR

[81] Girardi M.-Matzeu M., Essential critical points of linking type and solutions of minimal period to superquadratic hamiltonian systems, J. Nonlinear Analysis T.M.A., to appear. | Zbl

[82] Gordon W.B., Conservative dynamical systems involving strong forces, Trans. Amer. Math. Soc. 204 (1975), 113-135. | Zbl | MR

[83] Ghoussoub N.-Preiss D., A general mountain pass principle for locating and classifying critcal points, Annales Inst. H. Poincaré, Analyse Nonlineaire, 6 (1989), 321-330. | Zbl | Numdam

[84] Greco C., Periodic solutions of a class of singular Hamiltonian systems, J. Nonlinear Analysis T.M.A. 12 (1988), 259-270. | Zbl | MR

[85] Hofer H., A note on the topological degree at a critical point of mountain-pass type, Proc. Am. Math. Soc. 90 (1984), 309-315. | Zbl | MR

[86] Hofer H.-Wysocki K., First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems, Math. Ann. Vol. 288 (1990), 483-503. | Zbl | MR

[87] Hofer H.-Zehnder E., Periodic solutions on hypersurfaces and a result by C. Viterbo, Inv. Math. 90 (1987), 1-9. | Zbl | MR

[88] Klingenberg W., Lectures on closed geodesics, Springer, 1978. | Zbl | MR

[89] Kasdan J.L.-Warner F.W., Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math. 28 (1975), 567-597. | Zbl | MR

[90] Kovalevsky J., Introduction to celestial Mechanics, D. Reidel Publ. Co., Dordrecht, 1967. | Zbl

[91] Krasnoselskii M.A., Topological Methods in the theory of non-linear integral equations, Pergamon, Oxford, 1965.

[92] Landesman E.M.-Lazer A.C., Nonlinear perturbations of linear elliptic problems at resonance, J. Math. Mech. 19 (1970), 609-623. | Zbl | MR

[93] Lazer A.C.-Mckenna P.J., On the number of solutions of a nonlinear Dirichlet problem, J. Math. Anal. Appl. 84 (1981), 282-294. | Zbl | MR

[94] Lazzo M., Multiple positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, C.R. Acad. Sci. Paris, to appear. | Zbl

[95] Levi Civita T., Introduzione alla meccanica relativistica, Zanichelli, Bologna, 1928.

[96] Lusternik L.-Schnirelman L., Méthode topologique dans les problémes varationelles, Hermann, Paris (1934). | Zbl | JFM

[97] Majer P., Ljusternik-Schnirelman theory without Palais-Smale condition and singular dynamical systems, Annales Inst. H. Poincaré, Analyse Nonlin.8 (1991), 459-476. | Zbl | MR | Numdam

[98] Mawhin J.-Willem M., Critical point theory and hamiltonian systems, Springer, 1989. | Zbl | MR

[99] Moser J., Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Comm. Pure Appl. Math. 29 (1976), 727-747. | Zbl | MR

[100] Moser J., Regularization of Kepler's problem and the averaging method on a manifold, Comm. Pure Appl. Math. 23 (1970), 609-636. | Zbl | MR

[101] Ni W.M., On the existence of global vortex rings, J. d'Analyse Math. 37 (1980), 208-247. | Zbl | MR

[102] Nirenberg L., Variational and topological methods in nonlinear problems, Bull. A.M.S. 4-3 (1981), 267-302. | Zbl | MR

[103] Norbury J., A family of steady vortex rings, J. Fluid Mech. 57 (1973), 417-431. | Zbl

[104] Palais R., Lusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966), 115-132. | Zbl | MR

[105] Palais R.-Smale S., A generalized Morse theory, Bull. Amer. Math. Soc. 70 (1964), 165-171. | Zbl | MR

[106] Pohozaev S.I., Eigenfunctions of the equations Δu + λf(u) &#x003D 0, Soviet Math. 5 (1965), 1408-1411. | Zbl

[107] Rabinowitz P.H., Periodic solutions of hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), 157-184. | Zbl | MR

[108] Rabinowitz P.H., A variational method for finding periodic solutions of differential equations, Nonlinear Evolution Equations (M.G. Crandall ed.), Academic Press (1978), 225-251. | Zbl | MR

[109] Rabinowitz P.H., On a theorem of Hofer and Zehnder, Periodic solutions of hamiltonian systems and related topics (P.H. Rabinowitz et al. ed.), NATO ASI Series C Vol. 209, Reidel Publ. Co., 1986, 245-253. | Zbl | MR

[110] Rabinowitz P.H., Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. in Math. 65, A.M.S., Providence, 1986. | Zbl | MR

[111] Rabinowitz P.H., Homoclinic orbits for a class of Hamiltonian systems, Proceed. Royal Soc. Edinburgh, Vol.114A (1990), 33-38. | Zbl | MR

[112] Rabinowitz P.H.-Tanaka K., Some results on connecting orbits for a class of Hamiltonian systems, Math. Zeit., to appear. | Zbl

[113] Schwartz J.T., Generalizing the Lusternik-Schnirelman theory of critical points, Comm. Pure Appl. Math. 17 (1964), 307-315. | Zbl | MR

[114] Séré E., Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Zeit., to appear. | Zbl

[115] Serra E., Dynamical systems with singular potentials : existence and qualitative properties of periodic motions, Ph.D. Thesis, SISSA, Trieste (1991).

[116] Serra E.-Terracini S., Noncollision periodic solutions to some three- body like problems, Preprint SISSA, Trieste (1990).

[117] Srikanth P.N., Uniqueness of solutions of nonlinear Dirichlet problems, to appear.

[118] Stampacchia G., Le probléme de Dirichlet pour les équations elliptiques du second ordre a coefficients discontinuous, Ann. Inst. Fourier, 15 (1965), 189-258. | Zbl | MR | Numdam

[119] Struwe M., Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems, Manus. Math. 32 (1980), 335-364. | Zbl | MR

[120] Struwe M., Variational methods, Springer, 1990. | Zbl | MR

[121] Stuart C., Differential equations with discontinuos nonlinearities, Arch. Rat. Mech. Anal. 63 (1976), 59-75. | Zbl | MR

[122] Stuart C.-Toland J.F., A variational method for boundary value problems with discontinuous non-linearities, J. London Math. Soc. 21 (1980), 319-328. | Zbl | MR

[123] Szulkin A., Ljusternik-Schnirelmann theory on C1 manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), 119-139. | Zbl | MR | Numdam

[124] Tanaka K., Homoclinic orbits for a singular second order Hamiltonian system, Annales Inst. H. Poincaré, Analyse Non-lineaire, Vol.7 (1990), 427-438. | Zbl | MR | Numdam

[125] Tanaka K., Non-collision solutions for a second order singular Hamiltonian system with weak forces, Preprint (1991).

[126] Terracini S., Periodic solutions to dynamical systems with Keplerian type potentials, Ph.D. Thesis, SISSA, Trieste (1990).

[127] Van Groesen E.W.C., Analytical mini-max methods for Hamiltonian break orbits of prescribed energy, J. Math. Anal. Appl. 132 (1988), 1-12. | Zbl

[128] Wang Z.Q., On a superlinear elliptic equation, Annales Inst H. Poincaré Analyse nonlin. 8 (1991), 43-58. | Zbl | MR | Numdam

[129] Weinstein A., Normal modes for nonlinear hamiltonian systems, Invent. Math. 20 (1973), 45-57. | Zbl | MR

[130] Weinstein A., Periodic orbits for convex hamiltonian systems, Ann. of Math. 108 (1978), 507-518. | Zbl | MR

[131] Yang Janfu, Existence and asymptotic behaviour in planar vortex theory, to appear.

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