Uniqueness theorems through the method of moving spheres
Yanyan Li and Meijun Zhu
Source: Duke Math. J. Volume 80, Number 2
(1995), 383-417.
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Permanent link to this document: http://projecteuclid.org/euclid.dmj/1077246088
Mathematical Reviews number (MathSciNet): MR1369398
Zentralblatt MATH identifier: 0846.35050
Digital Object Identifier: doi:10.1215/S0012-7094-95-08016-8
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.
Mathematical Reviews (MathSciNet): MR95b:35019
Zentralblatt MATH: 0793.35034
[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.
Mathematical Reviews (MathSciNet): MR93a:35048
Zentralblatt MATH: 0784.35025
Digital Object Identifier: doi:10.1007/BF01244896
[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.
Mathematical Reviews (MathSciNet): MR95h:35053
Zentralblatt MATH: 0806.35129
Digital Object Identifier: doi:10.1002/cpa.3160470105
[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.
Mathematical Reviews (MathSciNet): MR92m:35084
Zentralblatt MATH: 0746.35006
Digital Object Identifier: doi:10.1080/03605309108820797
[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.
Mathematical Reviews (MathSciNet): MR90c:35075
Zentralblatt MATH: 0702.35085
Digital Object Identifier: doi:10.1002/cpa.3160420304
[6] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), no. 3, 615–622.
Mathematical Reviews (MathSciNet): MR93e:35009
Zentralblatt MATH: 0768.35025
Digital Object Identifier: doi:10.1215/S0012-7094-91-06325-8
Project Euclid: euclid.dmj/1077296071
[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.
Mathematical Reviews (MathSciNet): MR96j:35056
Zentralblatt MATH: 0830.35034
Digital Object Identifier: doi:10.1002/cpa.3160480606
[8] P. Cherrier, Problèmes de Neumann non linéaires sur les variétés riemanniennes, J. Funct. Anal. 57 (1984), no. 2, 154–206.
Mathematical Reviews (MathSciNet): MR86c:58154
Zentralblatt MATH: 0552.58032
Digital Object Identifier: doi:10.1016/0022-1236(84)90094-6
[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.
Mathematical Reviews (MathSciNet): MR94h:46052
Zentralblatt MATH: 0788.26015
Digital Object Identifier: doi:10.1112/jlms/s2-48.1.137
[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.
Mathematical Reviews (MathSciNet): MR95a:35038
Zentralblatt MATH: 0794.35049
Project Euclid: euclid.pjm/1102622458
[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.
Mathematical Reviews (MathSciNet): MR92f:58038
Zentralblatt MATH: 0713.53024
Digital Object Identifier: doi:10.1002/cpa.3160430703
[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.
Mathematical Reviews (MathSciNet): MR93e:53046
Zentralblatt MATH: 0766.53033
Digital Object Identifier: doi:10.2307/2946545
[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.
Mathematical Reviews (MathSciNet): MR80h:35043
Zentralblatt MATH: 0425.35020
Digital Object Identifier: doi:10.1007/BF01221125
Project Euclid: euclid.cmp/1103905359
[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.
Mathematical Reviews (MathSciNet): MR83f:35045
Zentralblatt MATH: 0465.35003
Digital Object Identifier: doi:10.1002/cpa.3160340406
[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.
Mathematical Reviews (MathSciNet): MR96e:35087
Zentralblatt MATH: 0827.35014
Digital Object Identifier: doi:10.1002/cpa.3160480502
[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.
Mathematical Reviews (MathSciNet): MR95a:35047
Zentralblatt MATH: 0820.35062
[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.
Mathematical Reviews (MathSciNet): MR92e:35024
Zentralblatt MATH: 0741.35014
Digital Object Identifier: doi:10.1080/03605309108820770
[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.
Mathematical Reviews (MathSciNet): MR94c:35017
Zentralblatt MATH: 0799.35068
Digital Object Identifier: doi:10.1002/cpa.3160460302
[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.
Mathematical Reviews (MathSciNet): MR46:2601
Zentralblatt MATH: 0236.53042
Project Euclid: euclid.jdg/1214430407
[25] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304–318.
Mathematical Reviews (MathSciNet): MR48:11545
Zentralblatt MATH: 0222.31007
Digital Object Identifier: doi:10.1007/BF00250468
[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.
Mathematical Reviews (MathSciNet): MR94e:53035
Zentralblatt MATH: 0733.53021
[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.
Mathematical Reviews (MathSciNet): MR96g:35079
Zentralblatt MATH: 0835.35055
[28] B. Ou, Positive harmonic functions on the upper half space satisfying a nonlinear boundary condition, Differential and Integral Equations, to appear.
Mathematical Reviews (MathSciNet): MR1392100
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