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On the spectral theory of positive operators and PDE applications
On gaussian curvature equation in $ \mathbb{R}^2 $ with prescribed nonpositive curvature
1. | Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China |
2. | Center for PDEs, School of Mathematical Sciences, East China Normal University, Shanghai Key Laboratory of PMMP, Shanghai 200062, China |
3. | IECL, UMR 7502, University of Lorraine, 57073 Metz, France |
$ \Delta u +K(x) e^{2u} = 0 \quad{\rm in}\;\; \mathbb{R}^2 $ |
$ K\le 0 $ |
$ \alpha_p(K) = \sup\left\{\alpha \in \mathbb{R}:\, \int_{ \mathbb{R}^2} |K(x)|^p(1+|x|)^{2\alpha p+2(p-1)} dx<+\infty\right\}, \quad \forall\; p \ge 1. $ |
$ ({\mathbb H}_1) $ |
$ \alpha_p(K)> -\infty $ |
$ p>1 $ |
$ \alpha_1(K) > 0 $ |
$ 0 < \alpha < \alpha_1(K) $ |
$ u_\alpha $ |
$ u_\alpha(x) = \alpha \ln |x|+ c_\alpha+o\big(|x|^{-\frac{2\beta}{1+2\beta}} \big) $ |
$ \beta\in (0, \, \alpha_1(K)-\alpha) $ |
$ K_0 \leq 0 $ |
$ \alpha_p(K_0) = -\infty $ |
$ p>1 $ |
$ \alpha_1(K_0) > 0 $ |
$ u_{\alpha_*} $ |
$ u_{\alpha_*} -\alpha_*\ln|x| = O(1) $ |
$ \alpha_* > 0 $ |
References:
[1] |
L. V. Ahlfors,
An extension of Schwartz’s lemma, Trans. Amer. Math. Soc., 43 (1938), 359-364.
doi: 10.2307/1990065. |
[2] |
K.-S. Cheng and C.-S. Lin,
Conformal metrics with prescribed nonpositive Gaussian on $ \mathbb{R}^2$, Calc. Var. PDE, 11 (2000), 203-231.
doi: 10.1007/s005260000037. |
[3] |
K.-S. Cheng and W.-M. Ni,
On the structure of the conformal Gaussian curvature equation on $ \mathbb{R}^2$, Duke Math. J., 62 (1991), 721-737.
doi: 10.1215/S0012-7094-91-06231-9. |
[4] |
K.-S. Cheng and W.-M. Ni,
On the structure of the conformal Gaussian curvature equation on $ \mathbb{R}^2$ Ⅱ, Math. Ann., 290 (1991), 671-680.
doi: 10.1007/BF01459266. |
[5] |
J. Kazdan and F. Warner,
Curvature functions for open 2-manifolds, Ann. Math., 99 (1974), 203-219.
doi: 10.2307/1970898. |
[6] |
R. McOwen,
On the equation $\Delta u+Ke^2u=f$ and prescribed negative curvature in $ \mathbb{R}^2$, J. Math. Anal. Appl., 103 (1984), 365-370.
doi: 10.1016/0022-247X(84)90133-1. |
[7] |
R. McOwen,
Conformal metrics in $ \mathbb{R}^2$ with prescribed Gaussian curvature and positive total curvature, Indiana Univ. Math. J., 34 (1985), 97-104.
doi: 10.1512/iumj.1985.34.34005. |
[8] |
W.-M. Ni,
On the elliptic equation $\Delta u+K(x)e^2u=0$ and conformal metric with prescribed Gaussian curvatures, Invent. Math., 66 (1982), 343-352.
doi: 10.1007/BF01389399. |
[9] |
D. Sattinger,
Conformal metrics in $ \mathbb{R}^2$ with prescribed curvature, Indiana Univ. Math. J., 22 (1972/73), 1-4.
doi: 10.1512/iumj.1973.22.22001. |
show all references
References:
[1] |
L. V. Ahlfors,
An extension of Schwartz’s lemma, Trans. Amer. Math. Soc., 43 (1938), 359-364.
doi: 10.2307/1990065. |
[2] |
K.-S. Cheng and C.-S. Lin,
Conformal metrics with prescribed nonpositive Gaussian on $ \mathbb{R}^2$, Calc. Var. PDE, 11 (2000), 203-231.
doi: 10.1007/s005260000037. |
[3] |
K.-S. Cheng and W.-M. Ni,
On the structure of the conformal Gaussian curvature equation on $ \mathbb{R}^2$, Duke Math. J., 62 (1991), 721-737.
doi: 10.1215/S0012-7094-91-06231-9. |
[4] |
K.-S. Cheng and W.-M. Ni,
On the structure of the conformal Gaussian curvature equation on $ \mathbb{R}^2$ Ⅱ, Math. Ann., 290 (1991), 671-680.
doi: 10.1007/BF01459266. |
[5] |
J. Kazdan and F. Warner,
Curvature functions for open 2-manifolds, Ann. Math., 99 (1974), 203-219.
doi: 10.2307/1970898. |
[6] |
R. McOwen,
On the equation $\Delta u+Ke^2u=f$ and prescribed negative curvature in $ \mathbb{R}^2$, J. Math. Anal. Appl., 103 (1984), 365-370.
doi: 10.1016/0022-247X(84)90133-1. |
[7] |
R. McOwen,
Conformal metrics in $ \mathbb{R}^2$ with prescribed Gaussian curvature and positive total curvature, Indiana Univ. Math. J., 34 (1985), 97-104.
doi: 10.1512/iumj.1985.34.34005. |
[8] |
W.-M. Ni,
On the elliptic equation $\Delta u+K(x)e^2u=0$ and conformal metric with prescribed Gaussian curvatures, Invent. Math., 66 (1982), 343-352.
doi: 10.1007/BF01389399. |
[9] |
D. Sattinger,
Conformal metrics in $ \mathbb{R}^2$ with prescribed curvature, Indiana Univ. Math. J., 22 (1972/73), 1-4.
doi: 10.1512/iumj.1973.22.22001. |
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