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On gaussian curvature equation in $ \mathbb{R}^2 $ with prescribed nonpositive curvature

  • * Corresponding author: Feng Zhou

    * Corresponding author: Feng Zhou

Dedicated to Professor Wei-Ming Ni’s seventieth birthday

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)

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  • 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.

    Mathematics Subject Classification: 35R06, 35A01, 35J66.

    Citation:

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    [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.
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    [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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