Skip to main content
Log in

On the existence of multiple, single-peaked solutions for a semilinear Neumann problem

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. Adams, R., Sobolev Spaces, Academic Press, 1975.

  2. Ambrosetti, A. & Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.

    Google Scholar 

  3. 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 

  4. 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 (1991), 79–93.

    Google Scholar 

  5. Berestycki, H., Gallouët, T. & Kavian, O., Equations de champs scalaires euclidiens nonlinéaires dans le plan, C. R. Acad. Sc. Paris, Série I Math. 297 (1983), 307–310.

    Google Scholar 

  6. Berestycki, H. & Lions, P.-L., Nonlinear scalar field equations I, existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), 313–375.

    Google Scholar 

  7. Coleman, S., Glaser, V. & Martin, A., Action minima among solutions to a class of Euclidean scalar field equations, Comm. Math. Phys. 58 (1978), 211–221.

    Google Scholar 

  8. Chow, S. N. & Hale, J. K., Methods of Bifurcation Theory, Springer-Verlag, 1982.

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

    Google Scholar 

  10. Ding, W.-Y., Positive solutions of Δu + u (N+2)/(N−2) = 0 on contractible domains, J. Part. Diff. Eqs. 2 (1989), 83–88.

    Google Scholar 

  11. Ding, W.-Y. & Ni, W.-M., On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rational Mech. Anal. 91 (1986), 283–308.

    Google Scholar 

  12. Fife, P. C., Semilinear elliptic boundary value problems with small parameters, Arch. Rational Mech. Anal. 52 (1973), 205–232.

    Google Scholar 

  13. Gierer, A. & Meinhardt, H., A theory of biological pattern formation, Kybernetik 12 (1972), 30–39.

    Google Scholar 

  14. Gidas, B., Ni, W.-M. & Nirenberg, L., Symmetry of positive solutions of nonlinear elliptic equations in R n, Advances in Math., Supplementary Studies 7A (1981), 369–402.

    Google Scholar 

  15. Gilbarg, D. & Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, Second Edition, Springer-Verlag, 1983.

  16. Keller, E. F. & Segel, L. A., Initiation of slime model aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399–415.

    Google Scholar 

  17. Kwong, M. K., Uniqueness of positive solutions Δu − u + u p = 0 in R n, Arch. Rational Mech. Anal. 105 (1989), 243–266.

    Google Scholar 

  18. Lions, P. L., The concentration-compactness principle in the calculus of variations. The locally compact case, Part 1 and Part 2, Ann. Inst. H. Poincaré, Anal. Nonlinéaire 1 (1984), 109–145, 223–283.

    Google Scholar 

  19. Lin, C.-S. & Ni, W.-M., On the diffusion coefficient of a semilinear Neumann problem, in Calculus of Variations and Partial Differential Equations, (S. Hildebrandt, D. Kinderlehrer & M. Miranda, Eds.), 160–174, Lecture Notes in Math. 1340, Springer-Verlag, 1988.

  20. Lin, C.-S., Ni, W.-M. & Takagi, I., Large amplitude stationary solutions to a chemotaxis system, J. Diff, Eqs. 72 (1988), 1–27.

    Google Scholar 

  21. McLeod, K. & Serrin, J., Uniqueness of positive radial solutions of Δu + f(u) = 0 in R n, Arch. Rational Mech. Anal. 99 (1987), 115–145.

    Google Scholar 

  22. Ni, W.-M. & Takagi, I., On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 45 (1991), 819–851.

    Google Scholar 

  23. Takagi, I., Point-condensation for a reaction-diffusion system, J. Diff. Eqs. 61 (1986), 208–249.

    Google Scholar 

  24. Wang, Z.-Q., On the existence of positive solutions for semilinear Neumann problems in exterior domains, to appear in Comm. Partial Diff. Eqs.

  25. Wang, Z.-Q., The effect of the domain geometry on the number of positive solutions of Neumann problems with critical exponents, preprint.

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by P. H. Rabinowitz

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, ZQ. On the existence of multiple, single-peaked solutions for a semilinear Neumann problem. Arch. Rational Mech. Anal. 120, 375–399 (1992). https://doi.org/10.1007/BF00380322

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00380322

Keywords

Navigation