Duke Mathematical Journal

Uniqueness theorems through the method of moving spheres

Yanyan Li and Meijun Zhu

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Source
Duke Math. J., Volume 80, Number 2 (1995), 383-417.

Dates
First available in Project Euclid: 19 February 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1077246088

Digital Object Identifier
doi:10.1215/S0012-7094-95-08016-8

Mathematical Reviews number (MathSciNet)
MR1369398

Zentralblatt MATH identifier
0846.35050

Subjects
Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 35B99: None of the above, but in this section 35K60: Nonlinear initial value problems for linear parabolic equations

Citation

Li, Yanyan; Zhu, Meijun. Uniqueness theorems through the method of moving spheres. Duke Math. J. 80 (1995), no. 2, 383--417. doi:10.1215/S0012-7094-95-08016-8. https://projecteuclid.org/euclid.dmj/1077246088


Export citation

References

  • [1] H. Berestycki, L. Caffarelli, and L. Nirenberg, Symmetry for elliptic equations in a half space, Boundary value problems for partial differential equations and applications ed. C. Baiocchi, RMA Res. Notes Appl. Math., vol. 29, Masson, Paris, 1993, pp. 27–42.
  • [2] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.) 22 (1991), no. 1, 1–37.
  • [3] H. Berestycki, L. Nirenberg, and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1994), no. 1, 47–92.
  • [4] H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of $-\Delta u=V(x)e\sp u$ in two dimensions, Comm. Partial Differential Equations 16 (1991), no. 8-9, 1223–1253.
  • [5] L. Caffarelli, B. Gidas, and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), no. 3, 271–297.
  • [6] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), no. 3, 615–622.
  • [7] W. Chen and C. Li, A necessary and sufficient condition for the Nirenberg problem, Comm. Pure Appl. Math. 48 (1995), no. 6, 657–667.
  • [8] P. Cherrier, Problèmes de Neumann non linéaires sur les variétés riemanniennes, J. Funct. Anal. 57 (1984), no. 2, 154–206.
  • [9] M. Chipot, I. Shafrir, and M. Fila, On the solutions to some elliptic equations with Neumann boundary conditions, preprint.
  • [10] K. S. Chou and C. W. Chu, On the best constant for a weighted Sobolev-Hardy inequality, J. London Math. Soc. (2) 48 (1993), no. 1, 137–151.
  • [11] K. S. Chou and T. Y. H. Wan, Asymptotic radial symmetry for solutions of $\Delta u+e\sp u=0$ in a punctured disc, Pacific J. Math. 163 (1994), no. 2, 269–276.
  • [12] J. F. Escobar, Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities, and an eigenvalue estimate, Comm. Pure Appl. Math. 43 (1990), no. 7, 857–883.
  • [13] J. F. Escobar, Conformal metrics with prescribed curvature on the boundary, preprint.
  • [14] J. F. Escobar, Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary, Ann. of Math. (2) 136 (1992), no. 1, 1–50.
  • [15] B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243.
  • [16] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), no. 4, 525–598.
  • [17] C. Gui, Symmetry of the blow-up set of a porous medium type equation, Comm. Pure Appl. Math. 48 (1995), no. 5, 471–500.
  • [18] B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition, Differential Integral Equations 7 (1994), no. 2, 301–313.
  • [19] C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Comm. Partial Differential Equations 16 (1991), no. 4-5, 585–615.
  • [20] Y. Y. Li, On $-\Delta u=K(x)u\sp 5$ in ${\bf R}\sp 3$, Comm. Pure Appl. Math. 46 (1993), no. 3, 303–340.
  • [21] Y. Y. Li, Prescribing scalar on $\mathbb{S}^{n}$ and related problems, part I, J. Differential Equations 120(195), 319–410. Prescribing scalar on $\mathbb{S}^{n}$ and related problems, Part II: Existence and compactness, preprint.
  • [22] P. Padilla, On some nonlinear elliptic equations, thesis, Courant Institute, 1994.
  • [23] J. Liouville, Sur l'équation aux différences partielles $(\parial^{2} \log \lambda/\parial u \partial v) \pm \lambda/2a^{2}=0$, J. de Math. 18 (1853), 71–72.
  • [24] M. Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geometry 6 (1971/72), 247–258.
  • [25] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304–318.
  • [26] R. Schoen, On the number of constant scalar curvature metrics in a conformal class, Differential geometry eds. H. B. Lawson, Jr. and K. Tenenblat, Pitman Monogr. Surveys Pure Appl. Math., vol. 52, Longman Sci. Tech., Harlow, 1991, pp. 311–320.
  • [27] S. Terracini, Symmetry properties of positive solutions to some elliptic equations with nonlinear boundary conditions, Differential Integral Equations 8 (1995), no. 8, 1911–1922.
  • [28] B. Ou, Positive harmonic functions on the upper half space satisfying a nonlinear boundary condition, Differential and Integral Equations, to appear.