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The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems

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References

  1. Bahri, A. & Coron, J. M., On a nonlinear elliptic equation involving the Sobolev exponent: the effect of the topology of the domain. Comm. Pure Appl. Math. 41 (1988), 253–294.

    Google Scholar 

  2. Benci, V. & Cerami, G., Positive solutions of some nonlinear elliptic problems in exterior domains. Arch. Rational Mech. Anal. 99 (1987), 283–300.

    Google Scholar 

  3. Benci, V. & Cerami, G., In preparation.

  4. Brezis, H. & Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36 (1983), 437–477.

    Google Scholar 

  5. Brezis, H. & Oswald, L., Remarks on sublinear elliptic equations, Nonlinear Analysis T.M.A. 10 (1986), 55–64.

    Google Scholar 

  6. Coron, J. M., Topologie et cas limite des injections de Sobolev. C. R. Ac. Se. Paris 299, Séries I (1984), 209–212.

    Google Scholar 

  7. Dancer, E. N., The effect of domain shape on the number of positive solutions of certain nonlinear equations. J. Diff. Equations 74 (1988), 120–156.

    Google Scholar 

  8. Dancer, E. N., A note on an equation with critical exponent. Bull. London Math. Soc. 20 (1988), 600–602.

    Google Scholar 

  9. Ding, W. Y., Positive solutions of Δu + uN+2 N−2=0 on contractible domains, preprint.

  10. Gidas, B., Ni, W. M. & Nirenberg, L., Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979), 209–243.

    Google Scholar 

  11. Gidas, B., Ni, W. M. & Nirenberg, L., Symmetry of positive solutions of nonlinear elliptic equations in RN. Mathematical Analysis and Applications: Part A. Advances in Mathematics Supplementary Studies. Vol. 7 A, 369–402.

  12. Krasnoselskii, M., Positive solutions of operator equations, Noordhoff, Groningen (1964).

    Google Scholar 

  13. Lions, P. L., The concentration-compactness principle in the calculus of variations. The limit case I–II. Revista Mat. Iberoamericana 1.1 (1985), 145–200 / 1.2 (1985), 45–121.

    Google Scholar 

  14. Ni, W. M. & Nussbaum, R. D., Uniqueness and nonuniqueness for positive radial solutions of Δu + f(u,r) = 0, Comm. Pure Appl. Math. 38 (1985).

  15. Passaseo, D., Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains. Manuscripta Math. 65 (1989), 147–166.

    Google Scholar 

  16. Strauss, W. A., Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149–162.

    Google Scholar 

  17. Struwe, M., A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z. 187 (1984), 511–517.

    Google Scholar 

  18. Rey, O., About a nonlinear equation involving the critical Sobolev exponent. Preprint.

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Communicated by H. Brezis

Dedicated to G. Prodi for his 65 th birthday

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Benci, V., Cerami, G. The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems. Arch. Rational Mech. Anal. 114, 79–93 (1991). https://doi.org/10.1007/BF00375686

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