A generalization of the saddle point method with applications
Martin Schechter
Annales Polonici Mathematici, Tome 57 (1992), p. 269-281 / Harvested from The Polish Digital Mathematics Library

We show that one can drop an important hypothesis of the saddle point theorem without affecting the result. We then show how this leads to stronger results in applications.

Publié le : 1992-01-01
EUDML-ID : urn:eudml:doc:275879
@article{bwmeta1.element.bwnjournal-article-apmv57z3p269bwm,
     author = {Martin Schechter},
     title = {A generalization of the saddle point method with applications},
     journal = {Annales Polonici Mathematici},
     volume = {57},
     year = {1992},
     pages = {269-281},
     zbl = {0780.35001},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv57z3p269bwm}
}
Martin Schechter. A generalization of the saddle point method with applications. Annales Polonici Mathematici, Tome 57 (1992) pp. 269-281. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv57z3p269bwm/

[000] [AH] H. Amann and P. Hess, A multiplicity result for a class of elliptic boundary value problems, Proc. Roy. Soc. Edinburgh 84A (1979), 145-151. | Zbl 0416.35029

[001] [AP] A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl. 93 (1973), 231-247. | Zbl 0288.35020

[002] [AR] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. | Zbl 0273.49063

[003] [BBF] P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with 'strong' resonance at infinity, Nonlinear Anal. 7 (1983), 981-1012. | Zbl 0522.58012

[004] [BF] H. Berestycki and D. G. de Figueiredo, Double resonance in semilinear elliptic problems, Comm. Partial Differential Equations 6 (1981), 91-120. | Zbl 0468.35043

[005] [BP] M. Berger and E. Podolak, On the solution of a nonlinear Dirichlet problem, Indiana Univ. Math. J. 24 (1975), 837-846. | Zbl 0329.35026

[006] [BS] M. S. Berger and M. Schechter, On the solvability of semilinear gradient operator equations, Adv. in Math. 25 (1977), 97-132. | Zbl 0354.47025

[007] [BN] H. Brezis and L. Nirenberg, Remarks on finding critical points, to appear. | Zbl 0751.58006

[008] [Cac1] N. P. Cac, On an elliptic boundary value problem at double resonance, J. Math. Anal. Appl. 132 (1988), 473-483. | Zbl 0682.35043

[009] [Cac2] N. P. Cac, On the number of solutions of an elliptic boundary value problem with jumping nonlinearity, Nonlinear Anal. 13 (1989), 341-351. | Zbl 0717.35031

[010] [Cac3] N. P. Cac, On nontrivial solutions of a Dirichlet problem whose jumping nonlinearity crosses a multiple eigenvalue, J. Differential Equations 80 (1989), 379-404. | Zbl 0713.35036

[011] [Cac4] N. P. Cac, On a boundary value problem with non-smooth jumping non-linearity, to appear.

[012] [Cas] A. Castro, Hammerstein integral equations with indefinite kernel, Internat. J. Math. Math. Sci. 1 (1978), 187-201. | Zbl 0391.45007

[013] [D] E. N. Dancer, Multiple solutions of asymptotically homogeneous problems, to appear. | Zbl 0850.35043

[014] [DF] D. G. de Figueiredo, Positive solutions of some classes of semilinear elliptic problems, in: Proc. Sympos. Pure Math. 45, Amer. Math. Soc., 1986, 371-379.

[015] [DFG] D. G. de Figueiredo et J. P. Gossez, Conditions de non-résonance pour certains problèmes elliptiques semi-linéaires, C. R. Acad. Sci. Paris 302 (1986), 543-545. | Zbl 0596.35049

[016] [DFLN] D. G. de Figueiredo, P. L. Lions and R. D. Nussbaum, A priori estimates and existence results for positive solutions of semilinear elliptic equations, J. Math. Pures Appl. 61 (1982), 41-63. | Zbl 0452.35030

[017] [GK] T. Gallouet et O. Kavian, Résultats d'existence et de non-existence pour certains problèmes demi linéaires d l'infini, Ann. Fac. Sci. Toulouse Math. 3 (1981), 201-246. | Zbl 0495.35001

[018] [H] P. Hess, On a nonlinear elliptic boundary value problem of the Ambrosetti-Prodi type, Boll. Un. Mat. Ital. 17A (1980), 189-192. | Zbl 0519.35028

[019] [KW] J. L. Kazdan and F. W. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math. 28 (1975), 567-597. | Zbl 0325.35038

[020] [LL] E. A. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1970), 609-623. | Zbl 0193.39203

[021] [La] A. C. Lazer, Introduction to multiplicity theory for boundary value problems with asymmetric nonlinearities, in: Partial Differential Equations, F. Cordoso et al. (eds.), Lecture Notes in Math. 1324, Springer, 1988, 137-165.

[022] [LM1] A. C. Lazer and P. J. McKenna, Multiplicity results for a class of semilinear elliptic and parabolic boundary value problems, J. Math. Anal. Appl. 107 (1985), 371-395. | Zbl 0584.35053

[023] [LM2] A. C. Lazer and P. J. McKenna, Critical point theory and boundary value problems with nonlinearities crossing multiple eigenvalues, I, II, Comm. Partial Differential Equations 10 (1985), 107-150; 11 (1986), 1653-1676. | Zbl 0572.35036

[024] [LM3] A. C. Lazer and P. J. McKenna, Multiplicity of solutions of nonlinear boundary value problems with nonlinearities crossing several hiqher eigenvalues, J. Reine Angew. Math. 368 (1986), 184-200. | Zbl 0588.35036

[025] [Lin] S. S. Lin, Some results for semilinear differential equations at resonance, J. Math. Anal. Appl. 93 (1983), 574-592. | Zbl 0531.47039

[026] [LM] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems I, Springer, Berlin 1972.

[027] [Lio] P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), 441-457.

[028] [MW] J. Mawhin and M. Willem, Critical points of convex perturbations of some indefinite quadratic forms and semilinear boundary value problems at resonance, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), 431-453. | Zbl 0678.35091

[029] [N1] L. Nirenberg, Variational and topological methods in nonlinear problems, Bull. Amer. Math. Soc. 4 (1981), 267-302. | Zbl 0468.47040

[030] [N2] L. Nirenberg, Variational methods in nonlinear problems, in: Lecture Notes in Math. 1365, Springer, 1988, 100-119.

[031] [P] E. Podolak, On the range of operator equations with an asymptotically nonlinear term, Indiana Univ. Math. J. 25 (1976), 1127-1137. | Zbl 0353.47035

[032] [Ra1] P. H. Rabinowitz, Variational methods for nonlinear eigenvalue problems, in: Eigenvalues of Nonlinear Problems, G. Prodi (ed.), C.I.M.E., Ed. Cremonese, Roma 1975, 141-195.

[033] [Ra2] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conf. Ser. in Math. 65, Amer. Math. Soc., 1986.

[034] [Ru] B. Ruf, On nonlinear elliptic boundary value problems with jumping nonlinearities, Ann. Mat. Pura Appl. 128 (1980), 133-151.

[035] [Sc1] M. Schechter, Nonlinear elliptic boundary value problems at strong resonance, Amer. J. Math. 112 (1990), 439-460. | Zbl 0721.35023

[036] [Sc2] M. Schechter, Solution of nonlinear problems at resonance, Indiana Univ. Math. J. 39 (1990), 1061-1080. | Zbl 0796.35058

[037] [Sc3] M. Schechter, A bounded mountain pass lemma without the (PS) condition and applications, Trans. Amer. Math. Soc. 331 (1992), 681-703. | Zbl 0757.35026

[038] [Sc4] M. Schechter, Nonlinear elliptic boundary value problems at resonance, Nonlinear Anal. 14 (1990), 889-903. | Zbl 0708.35033

[039] [Sc5] M. Schechter, A variation of the mountain pass lemma and applications, J. London Math. Soc. (2) 44 (1991), 491-502. | Zbl 0756.35032

[040] [Sc6] M. Schechter, The Hampwile theorem for nonlinear eigenvalues, Duke Math. J. 59 (1989), 325-335. | Zbl 0701.47036

[041] [So] S. Solimini, Some remarks on the number of solutions of some nonlinear elliptic problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), 143-156. | Zbl 0583.35044

[042] [Ta] E. Tarafdar, An approach to nonlinear elliptic boundary value problems, J. Austral. Math. Soc. 34 (1983), 316-335. | Zbl 0533.47054

[043] [Th1] K. Thews, A reduction method for some nonlinear Dirichlet problems, Nonlinear Anal. 3 (1979), 794-813.

[044] [Th2] K. Thews, Nontrivial solutions of elliptic equations at resonance, Proc. Roy Soc. Edinburgh 85A (1980), 119-129.