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Quasilinear elliptic eigenvalue problems

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Commentarii Mathematici Helvetici

Summary

The generalized Palais-Smale condition introduced in [26] is applied to obtain multiple solutions of variational eigen-value problems with quasilinear principal part, thereby extending some well-known existence results for semilinear elliptic problems.

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References

  1. Amann, H.,Lusternik-Schnirelman theory and non-linear eigenvalue problems, Math. Ann.199 (1972), 55–72.

    Article  MathSciNet  Google Scholar 

  2. Amann, H.,Existence and multiplicity theorems for semi-linear elliptic boundary value problems, Math. Z.151 (1976), 281–295.

    Article  MathSciNet  Google Scholar 

  3. Ambrosetti, A.,On the existence of multiple solutions for a class of nonlinear boundary value problems, Rend. Sem. Mat. Univ. Padova49 (1973), 195–204.

    MathSciNet  Google Scholar 

  4. Bahri, A.,Topological results on a certain class of functionals and applications, preprint.

  5. Bahri, A. andBerestycki, H.,A perturbation method in critical point theory and applications, to appear in Trans. AMS.

  6. Berger, M. S.,Multiple solutions of nonlinear operator equations arising from the calculus of variations, Proc. Symp. Pure Math. (1)18 (1969), 10–27.

    Google Scholar 

  7. Browder, F.,Existence theorems for nonlinear partial differential equations, Proc. Symp. Pure Appl. Math.16 (1968), 1–60.

    Google Scholar 

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

    Article  Google Scholar 

  9. Coffman, C. V.,A minimum-maximum principle for a class of nonlinear integral equations, J. Anal. Math.22 (1969), 391–419.

    MathSciNet  Google Scholar 

  10. Coffman, C. V.,On a class of nonlinear elliptic boundary value problems, J. Math. Mech.19 (1970), 351–356.

    MathSciNet  Google Scholar 

  11. Giaquinta, M.,Multiple integrals in the calculus of variations and nonlinear elliptic systems, SFB 72 Vorlesungsreihe6 (1981), Univ. of Bonn.

  12. Hempel, J. A.,Superlinear boundary value problems and non-uniqueness, Thesis, Univ. of New England, Australia (1970).

    Google Scholar 

  13. Hempel, J. A.,Multiple solutions for a class of nonlinear boundary value problems, Ind. Univ. Math. J.20 (1971), 983–996.

    Article  MathSciNet  Google Scholar 

  14. Krasnosels’kii, M. A.,Topological methods in the theory of nonlinear integral equations, Macmillan, N. Y., 1964.

  15. Ladyshenskaya, O. R. andUral’tseva, N. N. Linear and quasilinear elliptic equations, Acad. Press, New York (1968).

    Google Scholar 

  16. Lusternik, L. A. andSchnirelman, L.,Methodes topologiques dans les problèmes variationnels, Actual. Sci. Indust,188 (1934).

  17. Moser, J.,A new proof of De Giorgis’s theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math.13 (1960), 457–471.

    MathSciNet  Google Scholar 

  18. Nehari, Z.,Characteristic values associated with a class of nonlinear second order differential equations, Acta Math.105 (1961), 141–175.

    MathSciNet  Google Scholar 

  19. Palais, R. S.,Morse theory on Hilbert manifolds, Topology2 (1963), 299–340.

    Article  MathSciNet  Google Scholar 

  20. Pohožaev, S. I.,Eigenfunctions of the equation Δu+λf(u)=0, Doklady,165.1 (1965), 1408–1412.

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. Rabinowitz, P. H.,Some critical point theorems and applications to semilinear elliptic partial differential equations. Ann. Sc. Norm. Sup. Pisa (4)2 (1978), 215–223.

    Google Scholar 

  23. Schwartz, J. T.,Generalizing the Lusternik-Schnirelman theory of critical points, Comm. Pure Appl. Math.17 (1964), 307–315.

    MathSciNet  Google Scholar 

  24. Smale, S.,Morse theory and a nonlinear generalization of the Dirichlet problem, Annals Math.80 (1964), 382–396.

    Article  MathSciNet  Google Scholar 

  25. Struwe, M.,Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems, manusc. math.32 (1980), 335–364.

    Article  MathSciNet  Google Scholar 

  26. Struwe, M.,Multiple solutions of differential equations without the Palais-Smale condition, Math. Ann. 261 (1982), 399–412.

    Article  MathSciNet  Google Scholar 

  27. Wiegner, M.,On two-dimensional elliptic systems with a one-sided condition, Math. Z.178 (1981), 493–500.

    Article  MathSciNet  Google Scholar 

  28. Brezis, H.,Positive solutions of nonlinear elliptic equations in the case of critical Sobolev exponent, Nonlinear partial differential equations and their applications, Collège de France Sem., Vol. III (1982).

  29. Brezis, H. andNirenberg, L., Positive solutions of nonlinear elliptic equations involving critical sobolev exponents, preprint.

  30. Cerami, G., Fortunato, D., and Struwe, M., Bifurcation and multiplicity results for Yamabe’s equation, preprint.

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Authors and Affiliations

  1. Institut für Augewandte Mathematik, Universität Bonn, Benngstrasse 4-6, D-5300, Bonn 1

    Michael Struwe

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  1. Michael Struwe
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Additional information

This research was supported by the Sonderforschungsbereich 72 of the Deutsche Forschungsgemeinschaft.

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Struwe, M. Quasilinear elliptic eigenvalue problems. Commentarii Mathematici Helvetici 58, 509–527 (1983). https://doi.org/10.1007/BF02564649

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  • DOI: https://doi.org/10.1007/BF02564649

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