June  2020, 40(6): 3201-3214. doi: 10.3934/dcds.2020125

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

* Corresponding author: Feng Zhou

Dedicated to Professor Wei-Ming Ni’s seventieth birthday

Received  March 2019 Revised  January 2020 Published  February 2020

Fund Project: H.C. is supported by NSFC (No. 11726614 and 11661045). F.Z. and D.Y. are supported by Science and Technology Commission of Shanghai Municipality (STCSM), grant No. 18dz2271000. F.Z. is also supported by NSFC (No. 11726613 and 11431005)

The purpose of this paper is to study the solutions of
$ \Delta u +K(x) e^{2u} = 0 \quad{\rm in}\;\; \mathbb{R}^2 $
with
$ K\le 0 $
. We introduce the following quantities:
$ \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. $
Under the assumption
$ ({\mathbb H}_1) $
:
$ \alpha_p(K)> -\infty $
for some
$ p>1 $
and
$ \alpha_1(K) > 0 $
, we show that for any
$ 0 < \alpha < \alpha_1(K) $
, there is a unique solution
$ u_\alpha $
with
$ u_\alpha(x) = \alpha \ln |x|+ c_\alpha+o\big(|x|^{-\frac{2\beta}{1+2\beta}} \big) $
at infinity and
$ \beta\in (0, \, \alpha_1(K)-\alpha) $
. Furthermore, we show an example
$ K_0 \leq 0 $
such that
$ \alpha_p(K_0) = -\infty $
for any
$ p>1 $
and
$ \alpha_1(K_0) > 0 $
, for which we study the asymptotic behavior of solutions. In particular, we prove the existence of a solution
$ u_{\alpha_*} $
such that
$ u_{\alpha_*} -\alpha_*\ln|x| = O(1) $
at infinity for some
$ \alpha_* > 0 $
, which does not converge to a constant at infinity. This example exhibits a new phenomenon of solution with logarithmic growth, finite total curvature, and non-uniform asymptotic behavior at infinity.
Citation: Huyuan Chen, Dong Ye, Feng Zhou. On gaussian curvature equation in $ \mathbb{R}^2 $ with prescribed nonpositive curvature. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3201-3214. doi: 10.3934/dcds.2020125
References:
[1]

L. V. Ahlfors, An extension of Schwartz’s lemma, Trans. Amer. Math. Soc., 43 (1938), 359-364.  doi: 10.2307/1990065.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J. Kazdan and F. Warner, Curvature functions for open 2-manifolds, Ann. Math., 99 (1974), 203-219.  doi: 10.2307/1970898.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J. Kazdan and F. Warner, Curvature functions for open 2-manifolds, Ann. Math., 99 (1974), 203-219.  doi: 10.2307/1970898.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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