Advertisement

Uniqueness and nondegeneracy of positive radial solutions of \(\mathbf {div\,{\varvec{(}}{\varvec{\rho }} \nabla u{\varvec{)}} +{\varvec{\rho }}{\varvec{(}}-gu+hu^p{\varvec{)}}=0}\)

  • Naoki Shioji
  • Kohtaro Watanabe
Article

Abstract

We study the uniqueness and nondegeneracy of positive solutions of \(\mathrm {div}\,(\rho \nabla u) +\rho (-g u+h u^p)=0 \) in a ball, the entire space, an annulus, or an exterior domain under the Dirichlet boundary condition.

Mathematics Subject Classification

35A02 34B18 35B09 35J15 

Notes

Acknowledgments

The authors are grateful to the referee for his/her careful reading and invaluable comments.

References

  1. 1.
    Atkinson, F.V., Peletier, L.A.: Sur les solutions radiales de l’équation \(\Delta u+{\frac{1}{2} }x\cdot \nabla u+{\frac{1}{2}}\lambda u+\vert u\vert ^{p-1}u=0\). C. R. Acad. Sci. Paris Sér. I Math. 302, 99–101 (1986)Google Scholar
  2. 2.
    Bandle, C., Brillard, A., Flucher, M.: Green’s function, harmonic transplantation, and best Sobolev constant in spaces of constant curvature. Trans. Am. Math. Soc. 350, 1103–1128 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bandle, C., Peletier, L.A.: Best Sobolev constants and Emden equations for the critical exponent in \(S^3\). Math. Ann. 313, 83–93 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bandle, C., Benguria, R.: The Brézis-Nirenberg problem on \({\mathbb{S}^{3}}\). J. Differ. Equ. 178, 264–279 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brézis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88, 486–490 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brézis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36, 437–477 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brezis, H., Peletier, L.A.: Elliptic equations with critical exponent on spherical caps of \(S^ 3\). J. Anal. Math. 98, 279–316 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Byeon, J., Oshita, Y.: Uniqueness of standing waves for nonlinear Schrödinger equations. Proc. R. Soc. Edinb. Sect. A 138, 975–987 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, C.C., Lin, C.S.: Uniqueness of the ground state solutions of \(\Delta u+f(u)=0\), Commun. Partial Differ. Equ. 16, 1549–1572 (1991)Google Scholar
  10. 10.
    Chen, W., Wei, J.: On the Brezis-Nirenberg problem on \(\mathbf{S}^ 3\), and a conjecture of Bandle-Benguria, language=English, with English and French summaries. C. R. Math. Acad. Sci. Paris 341, 153–156 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Coffman, C.V.: Uniqueness of the ground state solution for \(\Delta u-u+u^{3}=0\) and a variational characterization of other solutions. Arch. Ration. Mech. Anal. 46, 81–95 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dohmen, C., Hirose, M.: Structure of positive radial solutions to the Haraux-Weissler equation. Nonlinear Anal. 33, 51–69 (1998)Google Scholar
  13. 13.
    Escobedo, M., Kavian, O.: Variational problems related to self-similar solutions of the heat equation. Nonlinear Anal. 11, 1103–1133 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Felmer, P., Martínez, S., Tanaka, K.: Uniqueness of radially symmetric positive solutions for \(-\Delta u+u=u^ p\) in an annulus. J. Differ. Equ. 245, 1198–1209 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hadj Selem, F.: Radial solutions with prescribed numbers of zeros for the nonlinear Schrödinger equation with harmonic potential. Nonlinearity 24, 1795–1819 (2011)Google Scholar
  16. 16.
    Hadj Selem, F., Kikuchi, H.: Existence and non-existence of solution for semilinear elliptic equation with harmonic potential and Sobolev critical/supercritical nonlinearities. J. Math. Anal. Appl. 387, 746–754 (2012)Google Scholar
  17. 17.
    Haraux, A., Weissler, F.B.: Nonuniqueness for a semilinear initial value problem, Indiana Univ. Math. J. 31, 167–189 (1982)Google Scholar
  18. 18.
    Hirose, M.: Structure of positive radial solutions to the Haraux-Weissler equation. II. Adv. Math. Sci. Appl. 9, 473–497 (1999)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Hirose, M.: Structure of positive radial solutions to scalar field equations with harmonic potential. J. Differ. Equ. 178, 519–540 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hirose, M., Ohta, M.: On positive solutions for nonlinear elliptic equations with harmonic potential (Bęedlewo/Warsaw, 2000). In: GAKUTO Internat. Ser. Math. Sci. Appl., vol. 17. Gakkōtosho, Tokyo, pp. 40–63 (2002)Google Scholar
  21. 21.
    Hirose, M.: Uniqueness of positive solutions to scalar field equations with harmonic potential. Funkcial. Ekvac. 50, 67–100 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kabeya, Y., Tanaka, K.: Uniqueness of positive radial solutions of semilinear elliptic equations in \(\mathbf{R}^ N\) and Séré’s non-degeneracy condition. Commun. Partial Differ. Equ. 24, 563–598 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kawano, N., Yanagida, E., Yotsutani, S.: Structure theorems for positive radial solutions to \({\rm div}(\vert Du\vert ^ {m-2}Du)+K(\vert x\vert )u^ q=0\) in \({\mathbf{R}}^ n\). J. Math. Soc. Jpn 45, 719–742 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kawano, N., Yanagida, E., Yotsutani, S.: Structure theorems for positive radial solutions to \(\Delta u+K(\vert x\vert )u^ p=0\) in \(\mathbf{R}^ n\). Funkcial. Ekvac. 36, 557–579 (1993)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Kwong, M.K.: Uniqueness of positive solutions of \(\Delta u-u+u^ p=0\), Arch. Ration. Mech. Anal. 105, 243–266 (1989)Google Scholar
  26. 26.
    Kwong, M.K., Li, Y.: Uniqueness of radial solutions of semilinear elliptic equations. Trans. Am. Math. Soc. 333, 339–363 (1992)Google Scholar
  27. 27.
    Li, Y., Ni, W.-M.: On conformal scalar curvature equations in \({\mathbf{R}}^ n\). Duke Math. J. 57, 895–924 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Li, Y., Ni, W.-M.: On the existence and symmetry properties of finite total mass solutions of the Matukuma equation, the Eddington equation and their generalizations. Arch. Ration. Mech. Anal. 108, 175–194 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Li, Y., Ni, W.-M.: On the asymptotic behavior and radial symmetry of positive solutions of semilinear elliptic equations in \({\mathbf{R}}^ n\). I. Asymptotic behavior. Arch. Ration. Mech. Anal. 118, 195–222 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Li, Y.: On the asymptotic behavior and radial symmetry of positive solutions of semilinear elliptic equations in \({\mathbf{R}}^ n\). II. Radial symmetry. Arch. Ration. Mech. Anal. 118, 223–243 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Li, Y.: On the positive solutions of the Matukuma equation. Duke Math. J. 70, 575–589 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    McLeod, K.: Uniqueness of positive radial solutions of \(\Delta u+f(u)=0\) in \({\mathbf{R}}^ n\). II. Trans. Am. Math. Soc. 339, 495–505 (1993)MathSciNetzbMATHGoogle Scholar
  33. 33.
    McLeod, K., Serrin, J.: Uniqueness of positive radial solutions of \(\Delta u+f(u)=0\) in \({\mathbf{R}}^ n\). Arch. Ration. Mech. Anal. 99, 115–145 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Naito, Y.: Non-uniqueness of solutions to the Cauchy problem for semilinear heat equations with singular initial data. Math. Ann. 329, 161–196 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Naito, Y.: Self-similar solutions for a semilinear heat equation with critical Sobolev exponent. Indiana Univ. Math. J. 57, 1283–1315 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Ni, W.-M.: Uniqueness of solutions of nonlinear Dirichlet problems. J. Differ. Equ. 50, 289–304 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Ni, W-M.: Uniqueness, nonuniqueness and related questions of nonlinear elliptic and parabolic equations. In: Nonlinear functional analysis and its applications, Part 2 (Berkeley, Calif., 1983), Proc. Sympos. Pure Math., vol. 45. Amer. Math. Soc. Providence, pp. 229–241 (1986)Google Scholar
  38. 38.
    Ni, W-M., Nussbaum, R.D.: Uniqueness and nonuniqueness for positive radial solutions of \(\Delta u+f(u,r)=0\). Commun. Pure Appl. Math. 38, 67–108 (1985)Google Scholar
  39. 39.
    Ni, W-M., Takagi, I.: Locating the peaks of least-energy solutions to a semilinear Neumann problem. Duke Math. J. 70(2), 247–281 (1993)Google Scholar
  40. 40.
    Ni, W.-M., Yotsutani, S.: Semilinear elliptic equations of Matukuma-type and related topics. Jpn. J. Appl. Math. 5, 1–32 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Noussair, E.S., Swanson, C.A.: Solutions of Matukuma’s equation with finite total mass. Indiana Univ. Math. J. 38, 557–561 (1989)Google Scholar
  42. 42.
    Peletier, L.A., Serrin, J.: Uniqueness of positive solutions of semilinear equations in \({\mathbf{R}}^{n}\). Arch. Ration. Mech. Anal. 81, 181–197 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Peletier, L.A., Terman, D., Weissler, F.B.: On the equation \(\Delta u+\frac{1}{2}x\cdot \nabla u+f(u)=0\). Arch. Ration. Mech. Anal. 94, 83–99 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Pucci, P., Serrin, J.: Uniqueness of ground states for quasilinear elliptic operators. Indiana Univ. Math. J. 47, 501–528 (1998)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Serrin, J., Tang, M.: Uniqueness of ground states for quasilinear elliptic equations. Indiana Univ. Math. J. 49, 897–923 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Sato, Y.: Multi-peak solutions for nonlinear Schrödinger equations. Thesis, Waseda University (2007)Google Scholar
  47. 47.
    Shioji, N., Watanabe, K.: A generalized Pohožaev identity and uniqueness of positive radial solutions of \(\Delta u+g(r)u+h(r)u^ p=0\). J. Differ. Equ. 255, 4448–4475 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Tang, M.: Uniqueness and global structure of positive radial solutions for quasilinear elliptic equations. Commun. Partial Differ. Equ. 26, 909–938 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Tang, M.: Uniqueness of positive radial solutions for \(\Delta u-u+u^ p=0\) on an annulus. J. Differ. Equ. 189, 148–160 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Weissler, F.B.: Asymptotic analysis of an ordinary differential equation and nonuniqueness for a semilinear partial differential equation. Arch. Ration. Mech. Anal. 91, 231–245 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Weissler, F.B.: Rapidly decaying solutions of an ordinary differential equation with applications to semilinear elliptic and parabolic partial differential equations. Arch. Ration. Mech. Anal. 91, 247–266 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Yanagida, E.: Structure of positive radial solutions of Matukuma’s equation. Jpn. J. Ind. Appl. Math. 8, 165–173 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Yanagida, E.: Uniqueness of positive radial solutions of \(\Delta u+g(r)u+h(r)u^ p=0\) in \({\mathbf{R}}^ n\). Arch. Ration. Mech. Anal. 115, 257–274 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Yanagida, E.: Uniqueness of positive radial solutions of \(\Delta u+f(u,\vert x\vert )=0\). Nonlinear Anal. 19, 1143–1154 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Yanagida, E., Yotsutani, S.: Classification of the structure of positive radial solutions to \(\Delta u+K(\vert x\vert )u^ p=0\) in \({\mathbf{R}}^n\). Arch. Ration. Mech. Anal. 124, 239–259 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Yanagida, E., Yotsutani, S.: Existence of positive radial solutions to \(\Delta u+K(\vert x\vert )u^ p=0\) in \({\mathbf{R}}^n\). J. Differ. Equ. 115, 477–502 (1995)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of EngineeringYokohama National UniversityYokohamaJapan
  2. 2.Department of Computer ScienceNational Defense AcademyYokosukaJapan

Personalised recommendations