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Differential equations with multiple solutions and nonlinear functional analysis

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References

  1. L. NIRENBERG, Variational and topological methods in nonlinear problems. Bull.A.M.S., 4-3 (1981), 267–302.

    Article  MathSciNet  MATH  Google Scholar 

  2. J. LERAY and J. SCHAUDER, Topologie et èquations fonctionelle. Ann. Sci. Ecole Norm. Sup., 51 (1934), 45–78.

    MathSciNet  MATH  Google Scholar 

  3. H.AMANN, Lectures on some fixed point theorems. Monog. de Matem., IMPA, Rio de Janeiro.

    Google Scholar 

  4. J.T. SCHWARTZ, Nonlinear Functional Analysis, Gordon&Breach, New York, 1969.

    MATH  Google Scholar 

  5. M.A. KRASNOSELSKII, Topological Methods in the theory of nonlinear integral equations. McMillan, New York, 1964.

    Google Scholar 

  6. P.H. RABINOWITZ, Some global results for nonlinear eigenvalue problems. J. Func. Anal., 7 (1971), 487–513.

    Article  MathSciNet  MATH  Google Scholar 

  7. R. PALAIS, Morse theory on Hilbert manifolds, Topology 2 (1963), 299–340.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. MARINO and G. PRODI, Metodi perturbativi nella teoria di Morse, Boll.U.M.I., 11 (1975), 1–32.

    MathSciNet  MATH  Google Scholar 

  9. E.H.SPANIER, Algebraic Topology, McGraw&Hill Co., 1966.

    Google Scholar 

  10. A.AMBROSETTI and D.LUPO, On a class of nonlinear Dirichlet problems with multiple solutions, Nonlin. Anal. TMA, to appear.

    Google Scholar 

  11. L.A.LUSTERNIK and L.G.SCHNIRELMAN, Topological methods in variational problems. Trudy Inst. Math.Mech., Moscow State Univ., (1930), 1–68.

    Google Scholar 

  12. F.E. BROWDER, Infinite dimensional manifolds and nonlinear ellptic eigenvalue problems. Ann. of Math., 82 (1965), 459–477.

    Article  MathSciNet  MATH  Google Scholar 

  13. R.S. PALAIS, Lusternik-Schnirelman theory on Banach manifolds, Topology, 5 (1966), 115–132.

    Article  MathSciNet  MATH  Google Scholar 

  14. J.T. SCHWARTZ, Generalizing the Lusternik-Schnirelman theory of critical points. Comm. Pure Appl. Math., 82 (1964), 307–315.

    Article  MathSciNet  MATH  Google Scholar 

  15. A. AMBROSETTI and P.H. RABINOWITZ, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973) 349–381.

    Article  MathSciNet  MATH  Google Scholar 

  16. I. EKELAND, Periodic solutions of Hamiltonian equations and a theorem of P.Rabinowitz, J. Diff. Eq., 34 (1979), 523–534.

    Article  MathSciNet  MATH  Google Scholar 

  17. H. BREZIS, J.M. CORON and L. NIRENBERG, Free vibrations for a nonlinear wave equations and a theorem of P.Rabinowitz. Comm. Pure Appl. Math., 33 (1980), 667–684.

    Article  MathSciNet  MATH  Google Scholar 

  18. W.M. NI, Some minimax principles and their applications in nonlinear elliptic equations. J. d'Analyse Math., 37 (1980), 248–275.

    Article  MathSciNet  MATH  Google Scholar 

  19. V. BENCI and P.H. RABINOWITZ, Critical point theorems for indefinite functionals. Invent.Math., 52 (1979), 241–273.

    Article  MathSciNet  MATH  Google Scholar 

  20. A.AMBROSETTI, Elliptic equations with jumping nonlinearities, J. Math. Phys. Sci., to appear.

    Google Scholar 

  21. S. COURANT and D. HILBERT, Methods of Mathematical Physics. Interscience, New York, 1965.

    MATH  Google Scholar 

  22. A. MANES and AM. MICHELETTI, Un'estensione della teoria variationale classica degli autovalori per operatori ellittici del secondo ordine. Boll. U.M.I. 7 (1973), 285–301.

    MathSciNet  MATH  Google Scholar 

  23. MG. KREIN and M.A. RUTMAN, Linear operators leaving invariant a cone in a Banach space. Am.Math.Soc.Transl. 1-10 (1950), 199–325.

    MathSciNet  MATH  Google Scholar 

  24. M.STRUWE, A note on a result of Ambrosetti and Mancini, Ann. Mat. Pura Appl., to appear.

    Google Scholar 

  25. A. AMBROSETTI and G. MANCINI, Sharp nonuniqueness results for some nonlinear problems. Nonlin. Anal.TMA 3-5 (1979), 635–645.

    Article  MathSciNet  MATH  Google Scholar 

  26. A. AMBROSETTI, On the existence of multiple solutions for a class of nonlinear boundary value problems. Rend.Sem.Mat.Univ.Padova, 49 (1973), 195–204.

    MathSciNet  MATH  Google Scholar 

  27. P.H. RABINOWITZ, Variational methods for nonlinear eigenvalue problems. Ind.Univ.Math.J. 23 (1974), 729–754.

    Article  MathSciNet  MATH  Google Scholar 

  28. D.C. CLARK, A variant of the Lusternik-Schnirelman theory. Ind.Univ.Math.J. 22 (1972), 65–74.

    Article  MathSciNet  MATH  Google Scholar 

  29. E.M. LANDESMAN and A.C. LAZER, Nonlinear perturbations of linear elliptic problems at resonance. J.Math.Mech. 19 (1970), 609–623.

    MathSciNet  MATH  Google Scholar 

  30. S. FUCIK, Solvability of nonlinear equations and boundary value problems. D.Reidel Publ. Co., Dordrecht, 1980.

    MATH  Google Scholar 

  31. H. AMANN,A. AMBROSETTI and G. MANCINI, Elliptic equations with noninvertible Fredholm linear part and bounded nonlinearities. Math.Zeit. 158 (1978), 179–194.

    Article  MathSciNet  MATH  Google Scholar 

  32. M. FITZPATRICK, Existence results for equations involving noncompact perturbations of Fredholm mappings with applications to differential equations. J.Math.Anal.Appl. 66 (1978), 151–177.

    Article  MathSciNet  MATH  Google Scholar 

  33. A. AMBROSETTI and P. HESS, Positive solutions of asymptotically linear ellptic eigenvalue problems. J.Math.Anal.Appl. 73 (1980), 411–422.

    Article  MathSciNet  MATH  Google Scholar 

  34. T.KATO and P.HESS, On some linear and nonlinear eigenvalue problems with an indefinite weight function. Comm.P.D.E. to appear.

    Google Scholar 

  35. P.HESS, On bifurcation from infinity for positive solutions of second order elliptic eigenvalue problems. To appear.

    Google Scholar 

  36. H.O.PEITGEN and K.SCHMITT, Global topological perturbation of nonlinear elliptic eigenvalue problems. To appear.

    Google Scholar 

  37. H.O.PEITGEN, J.SAUPE and K.SCHMITT, Nonlinear elliptic boundary value problems versus their finite difference approximations: Numerical irrilevant solutions. J.Reine Ang.Math., to appear

    Google Scholar 

  38. H. AMANN and P. HESS, A multiplicity result for a class of elliptic boundary value problems. Proc.Royal Soc. Ed. 84-A (1979), 145–151.

    Article  MathSciNet  MATH  Google Scholar 

  39. J.L. KASDAN and F.W. WARNER, Remarks on some quasilinear elliptic equations. Comm.Pure Appl.Math. 28 (1975), 567–597.

    Article  MathSciNet  Google Scholar 

  40. 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.

    Article  MathSciNet  MATH  Google Scholar 

  41. A.C. LAZER and P.J. MCKENNA, On the number of solutions of a nonlinear Dirichlet problem. J.Math.Anal.Appl. 84 (1981), 282–294.

    Article  MathSciNet  MATH  Google Scholar 

  42. S.SOLIMINI, Existence of a third solution for a class of bvp with jumping nonlinearities, to appear.

    Google Scholar 

  43. H.HOFER, Variational and topological methods in partially ordered Hilbert spaces. Math. Annalen, to appear.

    Google Scholar 

  44. E.N. DANCER, On the Dirichlet problem for weakly nonlinear elliptic partial differential equations. Proc. Royal Soc.Ed. 76-A (1977), 283–300.

    Article  MathSciNet  MATH  Google Scholar 

  45. B.RUF, On nonlinear elliptic problems with jumping nonlinearities. Ann.Mat. Pura Appl., to appear.

    Google Scholar 

  46. S.I. POHOZAEV, Eigenfunctions of the equation -Δu+λf(u)=0. Soviet Math. 5 (1965), 1408–1411.

    MathSciNet  Google Scholar 

  47. H.BREZIS and L.NIRENBERG, to appear.

    Google Scholar 

  48. A.AMBROSETTI, A perturbation theorem for superlinear boundary value problems. M.R.C. Univ. of Wisconsin — Madison, Tech.Summ. Report N. 1446, 1974.

    Google Scholar 

  49. A. BAHRI and H. BEERSTYCKI, A perturbation method in critical point theory and applications. Trans.A.M.S. 267-1 (1981), 1–32.

    Article  MathSciNet  Google Scholar 

  50. M. STRUWE, Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems. Manus. Math. 32 (1980), 335–364.

    Article  MathSciNet  MATH  Google Scholar 

  51. P.H.RABINOWITZ, Multiple critical points of perturbed symmetric functionals, to appear.

    Google Scholar 

  52. A. BAHRI, Topological results on a certain class of functionals and applications. J.Funct.Anal. 41 (1981), 397–427.

    Article  MathSciNet  MATH  Google Scholar 

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H. W. Knobloch Klaus Schmitt

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Ambrosetti, A. (1983). Differential equations with multiple solutions and nonlinear functional analysis. In: Knobloch, H.W., Schmitt, K. (eds) Equadiff 82. Lecture Notes in Mathematics, vol 1017. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0103232

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  • DOI: https://doi.org/10.1007/BFb0103232

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