Some variational theorems of mixed type and elliptic problems with jumping nonlinearities

Marino, Antonio; Saccon, Claudio

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 25 (1997) no. 3-4, pp. 631-665.

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

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

[3] A. Ambrosetti, Elliptic equations with jumping nonlinearities, J. Math and Physics 18 (1984), 1-12. | MR | Zbl

[4] A. Ambrosetti - G. Prodi, On the inversion of some differential mappings with singularities between Banach spaces, Annals of Mathematics 4 (93) (1972), 231-247. | MR | Zbl

[5] A. Ambrosetti - P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349-381. | MR | Zbl

[6] A. Bahri, Résolution générique d'une équation semilinéaire, C. R. Acad. de Sci. Paris Sér. A-B 291 1980. | MR | Zbl

[7] M. Berger - E. Podolac, On the solutions ofa nonlinear Dirichlet problem, Indiana Univ. Math. J. 24 (1975), 837-846. | Zbl

[8] C. Bertocchi - M. Degiovanni, On the existence of two branches of bifurcation for eigenvalue problems associated with variational inequalities, in Scritti in onore di Giovanni Melfi, volume 11 of Sci. Mat., pages 35-72. Vita e Pensiero, Milano, 1994. | MR | Zbl

[9] G. Bordigia, Problemi Perturbativi in Teoria dei punti Critici, Master's thesis, Università Cattolica del Sacro Cuore, Fac. S.M.F.N., Brescia, Feb. 1996.

[10] N.P. Cac, On trivial solutions of a Dirichlet problem whose jumping nonlinearity crosses a multiple eigenvalue, J. Diff. Equations 80 (1989), 379-404. | MR | Zbl

[11] N.P. Cac, On a boundary value problem with nonsmooth jumping nonlinearity, J. Diff. Equations 93 (1991), 238-259. | MR | Zbl

[12] K. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problems", volume 6 of Progress in Nonlinear Differential Equations and TheirApplications Birckhäuser, Boston, Basel, Berlin, 1991. | Zbl

[13] D.G. Costa - D.G. De Figueireido - P.N. Srikanth, The exact number of solutions for a class of ordinary differential equations through Morse index theory, J. Diff. Equations 96 (1992), 185-199. | MR | Zbl

[14] E.N. Dancer, On the Dirichlet problem for weak nonlinear elliptic partial differential equations, Proc. Roy. Soc. Edinburgh A 76 (1977), 283-300. | MR | Zbl

[15] E.N. Dancer, On the ranges of certain weakly nonlinear partial differential equations, Journal de Mathématiques Pures et Appliquées 57 (1978), 351-366. | MR | Zbl

[16] E.N. Dancer, Multiple solutions of asymptotically homogeneous problems, Annals of Mathematics f 57 (1988), 63-78. | MR

[17] E.N. Dancer, A counterexample to the Lazer-McKenna conjecture, Nonlinear Anal., Theory, Methods and Applications 13 (1989), 19-21. | MR | Zbl

[18] E.N. Dancer, Generic domain dependence for nonsmooth equations and the open set problem for jumping nonlinearities, Topological Meth. Nonlinear Anal. 1 (1993), 139-150. | MR | Zbl

[19] D.G. De Figueireido - J.P. Gossez, On the First Curve of the Fuik Spectrum of an Elliptic Operator, preprint, 1994. | MR

[20] S Fučik, Nonlinear equations with noninvertible linear part, Czechoslovak Math. J. 24 (1974), 467-495. | MR | Zbl

[21] S Fučik, Boundary value problems with jumping nonlinearities, Căsopis Ešt. Mat. 101 (1976), 69-87. | MR | Zbl

[22] T. Gallouet - O. Kavian, Résultats d' existence et de non existence de solutions pour certain problèmes demi-linéaire à l' infini, Ann. Fac. Sci. Toulouse 3 (1981), 201-246. | Numdam | MR | Zbl

[23] J.A. Hempel, Multiple solutions for a class of nonlinear boundary value problems, Indiana Univ. Math. J. 20 (1971), 983-996. | MR | Zbl

[24] H. Hofer, Variational and topological methods in partially ordered Hilbert spaces, Math. Ann. 261 (1982), 493-514. | MR | Zbl

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

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

[27] A.C. Lazer - P.J. M, On a conjecture related to the number of solutions of a nonlinear Dirichlet problem, Proc. Roy. Soc. Edinburgh 95 (1983), 275-283. | MR | Zbl

[28] A.C. Lazer - P.J. M, Critical point theory and boundary value problems with nonlinearities crossing multiple eigenvalues I, Comm. in Partial Differential Equations 10 (1985), 107-150. | MR | Zbl

[29] A.C. Lazer - P.J. M, Critical point theory and boundary value problems with nonlinearities crossing multiple eigenvalues II, Comm. in Partial Differential Equations 11 (1986), 1653-1676. | MR | Zbl

[30] J. Liu, Bifurcation for potential operators, Nonlinear Anal., Theory, Methods and Applications 4 (1990), 345-353. | MR | Zbl

[31] L. Lusternik - L. Schnirelmann, "Méthodes Topologiques dans le Problèmes Variationelles", Hermann, Paris, 1934 | Zbl

[32] C.A. Magalhaes, Semilinear elliptic problems with crossing of multiple eigenvalues, Comm. in Partial Differential Equations 15 (1990), 1265-1292. | MR | Zbl

[33] A. Marino - A. Micheletti - A. Pistoia, Some variational results on semilinear problems with asymptotically nonsymmetric behaviour, in Nonlinear Analysis, A Tribute in Honour of Giovanni Prodi, pages 243-256. S. N. S. Pisa, 1991. | MR | Zbl

[34] A. Marino - A. Micheletti - A. Pistoia, A nonsymmetric asymptotically linear elliptic problem, Topological Meth. Nonlinear Anal. 4 (1994), 289-339. | MR | Zbl

[35] A. Marino - A. Micheletti - A. Pistoia, Sul numero e anche sulla molteplicita delle soluzioni di alcuni problemi ellittici asintoticamente lineari e non simmetrici, in Proceedings of the congress: "Variational Methods in Nonlinear Analysis and Differential Equations" held in Genova, June 1993, pages 123-141. E.C.I.G., 1994.

[36] A. Marino - A. Micheletti - A. Pistoia, On some elliptic asymptotically linear and nonsymmetric problems, Atti Sem. Mat. Fis. Univ. Modena XLIII (1995), 401-423. | MR | Zbl

[37] A. Marino - D. Passaseo, A jumping behaviour induced by an obstacle, in M. GIRARDI - M. MATZEU - F. PACELLA (Editors) Progress in Variational Methods in Hamiltonian Systems and Elliptic Equations, pages 127-143. Pitman, 1992. | MR | Zbl

[38] A. Marino - C. Saccon, Alcuni teoremi variazionali di tipo misto e un problema ellittico asintoticamente asimmetrico, Atti Sem. Mat. Fis. Univ. Modena, to appear.

[39] A. Micheletti, A remark on the resonance set for a semilinear elliptic equation, Proc. Roy. Soc. Edinburgh A 124 (1994), 803-809. | MR | Zbl

[40] A. Micheletti - A. Pistoia, A variational viewpoint on the existence of three solutions for some elliptic problems, Differential and Integral Equations 6 (1995), 67-80. | MR

[41] A. Micheletti - A. Pistoia, A note on the resonance set for a semilinear elliptic equation and an application to jumping nonlinearities, Topological Meth. Nonlinear Anal. 9 (5) (1996), 1029-1042. | MR

[42] M. Morse - S.S. Cairns, "Critical Point Theory in Global Analysis and Differential Topology ", Academic Press, New York, 1969. | MR | Zbl

[43] R. Palais, Morse theory on Hilbert manifolds, Topology 2 (1963), 299-340. | MR | Zbl

[44] P.H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differentia Equations " volume 65 of Regional Conference Series in Mathematics, American Matehmatical Society, Providence, R. I., 1986. | MR | Zbl

[45] M. Ramos, A critical point theorem suggested by an elliptic problem with asymmetric nonlinearities, preprint. | MR

[46] B. Ruf, On linear elliptic problems with jumping nonlinearities, Annals of Mathematics 4 (1981), 133-151. | MR | Zbl

[47] B. Ruf, Remarks and generalizations related to a recent result ofA, Lazer and P. J. McKenna Nonlinear Anal., Theory, Methods and Applications 9 (1985), 1325-1330. | MR | Zbl

[48] M. Schechter - K. Tintarev, Pairs of critical points produced by linking subsets with application to semilinear elliptic problems, Bull. Soc. Math. Belg. B 44 (1992), 249-261. | MR | Zbl

[49] S. Solimini, Existence of a third solution for a class of b. p. v. with jumping nonlinearities, Nonlinear Anal., Theory, Methods and Applications 7 (1983), 917-927. | MR | Zbl

[50] S. Solimini, Some remarks on the number of solutions of some semilinear elliptic problems, Ann. Inst. Henri Poincaré 2 (1985), 143-156. | Numdam | MR | Zbl