Skip to main content

Some Aspects of Semilinear Elliptic Equations on ℝn

  • Conference paper

Part of the book series: Mathematical Sciences Research Institute Publications ((MSRI,volume 13))

Abstract

In this expository paper, I intend to give a partial survey of semilinear elliptic equations on n.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. L. V. Ahlfors, An extension of Schwarz’s lemma, Trans. Amer. Math. Soc. 43 (1938), 359–364.

    MathSciNet  Google Scholar 

  2. L. V. Ahlfors and L. Bers, Riemann’s mapping theorem for variable metrics, Ann. Math. 72 (1960), 385–404.

    Article  MATH  MathSciNet  Google Scholar 

  3. K. Ako and T. Kusano, On bounded solutions of second order elliptic differential equations, J. Fac. Sci. Univ. Tokyo (Sect. I) 11 (1964), 29–37.

    MATH  MathSciNet  Google Scholar 

  4. P. Aviles and R. McOwen, Conformal deformations of complete manifolds with negative curvature, J. Diff. Geom. 21 (1985), 269–281.

    MATH  MathSciNet  Google Scholar 

  5. J. Batt, W. Faltenbacher, and E. Horst, Stationary spherically symmetric models in stellar dynamics, Arch. Rat. Mech. Anal. 93 (1986), 159–183.

    Article  MATH  MathSciNet  Google Scholar 

  6. H. Berestycki and P. L. Lions, Nonlinear scalar field equations I, Arch. Rat. Mech. Anal. 82 (1983), 313–345.

    MATH  MathSciNet  Google Scholar 

  7. H. Berestycki, P. L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in n, Indiana Univ. Math. J. 30 (1981), 141–157.

    Article  MATH  MathSciNet  Google Scholar 

  8. M. S. Berger, Onthe existence and structure of stationary states for a nonlinear Klein-Gordon equation, J. Funct. Anal. 9 (1972), 249–261.

    Article  MATH  Google Scholar 

  9. E. Calabi, An extension of E. Hopf maximum principle with an application to Riemannian geometry, Duke Math. J. 25 (1958), 45–56.

    Article  MATH  MathSciNet  Google Scholar 

  10. C.-Y. Chen, K.-S. Cheng, and W.-N. Yu, Conformal deformations of metrics on H n (-1) with prescribed scalar curvatures, preprint.

    Google Scholar 

  11. K.-S. Cheng and J. T. Lin, Onthe elliptic equations Δu = K(x)u σ and Δu = K(x)e 2u, Trans. Amer. Math. Soc, to appear.

    Google Scholar 

  12. K.-S. Cheng, F.-S. P. Tsen, and W.-N. Yu, Conformal deformations of metrics on Poincare disk, preprint.

    Google Scholar 

  13. C. V. Coffman, Uniqueness of the ground state solution for Δu = -u + u3 = 0 and a variational characterization of other solutions, Arch. Rat. Mech. Anal. 46 (1972), 81–95.

    Article  MATH  MathSciNet  Google Scholar 

  14. W.-Y. Ding, On a conformally invariant elliptic equation onn, Comm. Math. Phys. 107 (1986), 331–335.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. W.-Y. Ding and W.-M. Ni, On the elliptic equation Δu+ Ku (n+2)/(n-2) = 0 and related topics, Duke Math. J. 52 (1985), 485–506.

    Article  MATH  MathSciNet  Google Scholar 

  17. A. S. Eddington, The dynamics of a globular stellar system, Monthly Notices of the Royal Astronomical Society 75 (1915), 366–376.

    MATH  Google Scholar 

  18. A. Friedman, Bounded entire solutions of elliptic equations, Pacific J. Math. 44 (1973), 497–507.

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  21. H. G. Kaper and M.-K. Kwong, Uniqueness of nonnegative solutions of a class of semilinear elliptic equations, preprint.

    Google Scholar 

  22. N. Kawano, On bounded entire solutions of semilinear elliptic equations, Hiroshima Math. J. 14 (1984), 125–158.

    MATH  MathSciNet  Google Scholar 

  23. J. Kazdan, Prescribing the Curvature of a Riemannian Manifold, NSF-CBMS Regional Conference Lecture Notes 57 (1985).

    Google Scholar 

  24. C. E. Kenig and W.-M. Ni, An exterior Dirichlet problem with applications to some nonlinear equations arising in geometry, Amer. J. Math. 106 (1984), 689–702.

    Article  MATH  MathSciNet  Google Scholar 

  25. C. E. Kenig and W.-M. Ni, On the elliptic equation Lu - k + Ke 2u = 0, Ann. Scuo. Norm. Sup. Pisa (Series IV) 12 (1985), 191–224.

    MATH  MathSciNet  Google Scholar 

  26. T. Kusano and M. Naito, Oscillation theory of entire solutions of second order superlinear elliptic equations, Funkcial Ekvac, to appear.

    Google Scholar 

  27. Y. Li, Remarks on a semilinear elliptic equation in n, to appear in J. Diff. Eqns.

    Google Scholar 

  28. Y. Li and W.-M. Ni, in preparation.

    Google Scholar 

  29. P. L. Lions, The concentration-compactness principle in the calculus of variations, the limit case, Revista Mat. Iberoamericana 1 (1985), 145–201; II, 45–121.

    Google Scholar 

  30. C.-S. Lin and S.-S. Lin, in preparation.

    Google Scholar 

  31. C.-S. Lin and W.-M. Ni, A counterexample to the Nodal Domain Conjecture and a related semilinear equation, Proc. Amer. Math. Soc, to appear..

    Google Scholar 

  32. F.-H. Lin, On the elliptic equation D i [A ij (x)D j U] -k(x)U + K(x)U p = 0, Proc. Amer. Math. Soc. 95 (1985), 219–226.

    MATH  MathSciNet  Google Scholar 

  33. F.-H. Lin and W.-M. Ni, On the least growth of harmonic functions and the boundary behaviour of Riemann mappings, Comm. PDE 10 (1985), 767–786.

    Article  MATH  MathSciNet  Google Scholar 

  34. W. Littman, G. Stampacchia, and H. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuo. Norm. Sup. Pisa (Series III) 17 (1963), 45–79.

    MathSciNet  Google Scholar 

  35. C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal and projective transformations, in “Contributions to analysis,” Academic Press, 1974, pp. 245–272.

    Google Scholar 

  36. K. McLeod and J. Serrin, Uniqueness of positive radial solutions of Δu + f (u) = 0 in ℝn, Arch. Rat. Mech. Anal., to appear.

    Google Scholar 

  37. McLeod, J.B., W.-M. Ni, and J. Serrin, in preparation.

    Google Scholar 

  38. R. McOwen, The behavior of the Laplacian on weighted Sobolev spaces, Comm. Pure Appl. Math. 32 (1979), 783–795.

    Article  MATH  MathSciNet  Google Scholar 

  39. R. McOwen, On the equation Δu + Ke 2u = f and prescribed negative curvature in 2, J. Math. Anal. Appl. 103 (1984), 365–370.

    Article  MATH  MathSciNet  Google Scholar 

  40. R. McOwen, Conformai metric in2 with prescribed Gaussian curvature and positive total curvature, Indiana Univ. Math. J. 34 (1985), 97–104.

    Article  MathSciNet  Google Scholar 

  41. J. Moser, On a nonlinear problem in differential geometry, in “Dynamical systems,” M. Peixoto, ed., Academic Press, New York, 1973.

    Google Scholar 

  42. M. Naito, A note on bounded positive entire solutions of semilinear elliptic equations, Hiroshima Math. J. 14 (1984), 211–214.

    MATH  MathSciNet  Google Scholar 

  43. Z. Nehari, On a nonlinear differential equation arising in nuclear physics, Proc. Royal Irish Acad. 62 (1963), 117–135.

    MATH  MathSciNet  Google Scholar 

  44. W.-M. Ni, On the elliptic equation Δu + K(x)u(n+2)/(n-2) = 0, its generalization and applications in geometry, Indiana Univ. Math. J. 31 (1982), 493–529.

    Article  MATH  MathSciNet  Google Scholar 

  45. W.-M. Ni, On the elliptic equation Δu+Ke 2u = 0 and conformed metrics with prescribed Gaussian curvatures, Invent. Math. 66 (1982), 343–352.

    Article  MATH  MathSciNet  Google Scholar 

  46. W.-M. Ni and J. Serrin, Nonexistence theorems for quasilinear partial differential equations, Rend. Circolo Mat. Palermo. (Centenary Supplement), Series II 8 (1985), 171–185.

    MATH  MathSciNet  Google Scholar 

  47. W.-M. Ni and J. Serrin, Existence and nonexistence theorems for ground states for quasilinear partial differential equations, Accad. Naz. dei Lincei 77 (1986), 231–257.

    Google Scholar 

  48. W.-M. Ni and J. Serrin, Nonexistence theorems for singular solutions of quasilinear partial differential equations, Comm. Pure Appl. Math. 39 (1986), 379–399.

    Article  MATH  MathSciNet  Google Scholar 

  49. W.-M. Ni and S. Yotsutani, On Matukuma’s equation and related topics, Proc. Japan Acad. (Series A) 62 (1986), 260–263.

    MATH  MathSciNet  Google Scholar 

  50. W.-M. Ni and S. Yotsutani, Semilinear elliptic equations of Matukuma-type and related topics, Japan J. Appl. Math., to appear.

    Google Scholar 

  51. L. Nirenberg and H. F. Walker, The null spaces of elliptic partial differential operators in n, J. Math. Anal. Appl. 42 (1973), 271–301.

    Article  MATH  MathSciNet  Google Scholar 

  52. E. S. Noussair and C. A. Swanson, Existence theorems for generalized Klein-Gordon equations, Bull. Amer. Math. Soc. (New Series) 8 (1983), 333–336.

    Article  MATH  MathSciNet  Google Scholar 

  53. O. A. Oleinik, On the equation Δu + k(x)e u = 0, Russian Math. Surveys 33 (1978), 243–244.

    Article  Google Scholar 

  54. R. Osserman, On the inequality Δu > f(u), Pacific J. Math. 7 (1957), 1641–1647.

    MATH  MathSciNet  Google Scholar 

  55. L. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in ℝ n, Arch. Rat. Mech. Anal. 81 (1983), 181–197.

    Article  MATH  MathSciNet  Google Scholar 

  56. P. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (1973), 161–202.

    Article  MATH  MathSciNet  Google Scholar 

  57. D. H. Sattinger, Conformai metrics in 2 with prescribed curvature, Indiana Univ. Math. J. 22 (1972), 1–4.

    Article  MATH  MathSciNet  Google Scholar 

  58. R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geom. 20 (1984), 479–495.

    MATH  MathSciNet  Google Scholar 

  59. R. Schoen, The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation, preprint.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  61. C. A. Stuart, Bifurcation for Dirichlet problems without eigenvalues, Proc. London Math. Soc. (3) 45 (1982), 169–192.

    Article  MATH  MathSciNet  Google Scholar 

  62. H. Wittich, Ganze Losungen der Differentialgleichung Δu = e u, Math. Z. 49 (1944), 579–582.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1988 Springer-Verlag New York Inc.

About this paper

Cite this paper

Ni, WM. (1988). Some Aspects of Semilinear Elliptic Equations on ℝn . In: Ni, WM., Peletier, L.A., Serrin, J. (eds) Nonlinear Diffusion Equations and Their Equilibrium States II. Mathematical Sciences Research Institute Publications, vol 13. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9608-6_10

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-9608-6_10

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-9610-9

  • Online ISBN: 978-1-4613-9608-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics