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1992 | 57 | 3 | 269-281
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A generalization of the saddle point method with applications

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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.
  • Department of Mathematics, University of California, Irvine, California 92717, U.S.A.
  • [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.
  • [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.
  • [AR] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381.
  • [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.
  • [BF] H. Berestycki and D. G. de Figueiredo, Double resonance in semilinear elliptic problems, Comm. Partial Differential Equations 6 (1981), 91-120.
  • [BP] M. Berger and E. Podolak, On the solution of a nonlinear Dirichlet problem, Indiana Univ. Math. J. 24 (1975), 837-846.
  • [BS] M. S. Berger and M. Schechter, On the solvability of semilinear gradient operator equations, Adv. in Math. 25 (1977), 97-132.
  • [BN] H. Brezis and L. Nirenberg, Remarks on finding critical points, to appear.
  • [Cac1] N. P. Cac, On an elliptic boundary value problem at double resonance, J. Math. Anal. Appl. 132 (1988), 473-483.
  • [Cac2] N. P. Cac, On the number of solutions of an elliptic boundary value problem with jumping nonlinearity, Nonlinear Anal. 13 (1989), 341-351.
  • [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.
  • [Cac4] N. P. Cac, On a boundary value problem with non-smooth jumping non-linearity, to appear.
  • [Cas] A. Castro, Hammerstein integral equations with indefinite kernel, Internat. J. Math. Math. Sci. 1 (1978), 187-201.
  • [D] E. N. Dancer, Multiple solutions of asymptotically homogeneous problems, to appear.
  • [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.
  • [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.
  • [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.
  • [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.
  • [H] P. Hess, On a nonlinear elliptic boundary value problem of the Ambrosetti-Prodi type, Boll. Un. Mat. Ital. 17A (1980), 189-192.
  • [KW] J. L. Kazdan and F. W. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math. 28 (1975), 567-597.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [Lin] S. S. Lin, Some results for semilinear differential equations at resonance, J. Math. Anal. Appl. 93 (1983), 574-592.
  • [LM] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems I, Springer, Berlin 1972.
  • [Lio] P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), 441-457.
  • [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.
  • [N1] L. Nirenberg, Variational and topological methods in nonlinear problems, Bull. Amer. Math. Soc. 4 (1981), 267-302.
  • [N2] L. Nirenberg, Variational methods in nonlinear problems, in: Lecture Notes in Math. 1365, Springer, 1988, 100-119.
  • [P] E. Podolak, On the range of operator equations with an asymptotically nonlinear term, Indiana Univ. Math. J. 25 (1976), 1127-1137.
  • [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.
  • [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.
  • [Ru] B. Ruf, On nonlinear elliptic boundary value problems with jumping nonlinearities, Ann. Mat. Pura Appl. 128 (1980), 133-151.
  • [Sc1] M. Schechter, Nonlinear elliptic boundary value problems at strong resonance, Amer. J. Math. 112 (1990), 439-460.
  • [Sc2] M. Schechter, Solution of nonlinear problems at resonance, Indiana Univ. Math. J. 39 (1990), 1061-1080.
  • [Sc3] M. Schechter, A bounded mountain pass lemma without the (PS) condition and applications, Trans. Amer. Math. Soc. 331 (1992), 681-703.
  • [Sc4] M. Schechter, Nonlinear elliptic boundary value problems at resonance, Nonlinear Anal. 14 (1990), 889-903.
  • [Sc5] M. Schechter, A variation of the mountain pass lemma and applications, J. London Math. Soc. (2) 44 (1991), 491-502.
  • [Sc6] M. Schechter, The Hampwile theorem for nonlinear eigenvalues, Duke Math. J. 59 (1989), 325-335.
  • [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.
  • [Ta] E. Tarafdar, An approach to nonlinear elliptic boundary value problems, J. Austral. Math. Soc. 34 (1983), 316-335.
  • [Th1] K. Thews, A reduction method for some nonlinear Dirichlet problems, Nonlinear Anal. 3 (1979), 794-813.
  • [Th2] K. Thews, Nontrivial solutions of elliptic equations at resonance, Proc. Roy Soc. Edinburgh 85A (1980), 119-129.
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