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Volume 4, Issue 2
Solutions of Elliptic Equations ΔU+K(x)e2u=f(x)

Pan Xiugbin

J. Part. Diff. Eq., 4 (1991), pp. 36-44.

Published online: 1991-04

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  • Abstract
In this paper we consider the elliptic equation Δu + K(x)e^{2u} = f(x), which arises from prescribed curvature problem in Riemannian geometry. It is proved that if K(x) is negative and continuous in R², then for any f ∈ L²_{loc} (R²) such that f(x) ≤ K(x), the equation possesses a positive solution. A uniqueness theorem is also given.
  • Keywords

elliptic equations prescribed curvature problem monotone operators Kato's inequality

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COPYRIGHT: © Global Science Press

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@Article{JPDE-4-36, author = {}, title = {Solutions of Elliptic Equations ΔU+K(x)e2u=f(x)}, journal = {Journal of Partial Differential Equations}, year = {1991}, volume = {4}, number = {2}, pages = {36--44}, abstract = { In this paper we consider the elliptic equation Δu + K(x)e^{2u} = f(x), which arises from prescribed curvature problem in Riemannian geometry. It is proved that if K(x) is negative and continuous in R², then for any f ∈ L²_{loc} (R²) such that f(x) ≤ K(x), the equation possesses a positive solution. A uniqueness theorem is also given.}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5766.html} }
TY - JOUR T1 - Solutions of Elliptic Equations ΔU+K(x)e2u=f(x) JO - Journal of Partial Differential Equations VL - 2 SP - 36 EP - 44 PY - 1991 DA - 1991/04 SN - 4 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jpde/5766.html KW - elliptic equations KW - prescribed curvature problem KW - monotone operators KW - Kato's inequality AB - In this paper we consider the elliptic equation Δu + K(x)e^{2u} = f(x), which arises from prescribed curvature problem in Riemannian geometry. It is proved that if K(x) is negative and continuous in R², then for any f ∈ L²_{loc} (R²) such that f(x) ≤ K(x), the equation possesses a positive solution. A uniqueness theorem is also given.
Pan Xiugbin. (1970). Solutions of Elliptic Equations ΔU+K(x)e2u=f(x). Journal of Partial Differential Equations. 4 (2). 36-44. doi:
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