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In this paper, we consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a complete Riemannian manifold: $$\Delta_{V}u+cu^{\alpha}=0,$$ where $c$, $\alpha$ are two real constants and $c\neq0$. By applying Bochner formula and the maximum principle, we obtain local gradient estimates for positive solutions of the above equation on complete Riemannian manifolds with Bakry-\'{E}mery Ricci curvature bounded from below, which generalize some results of \cite{MHL2017}.
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참고문헌 (12)
참고문헌 신청
Kevin Brighton / 2013 / A Liouville-Type Theorem for Smooth Metric Measure Spaces / Journal of Geometric Analysis 23(2):562~570
Qun Chen / 2012 / Existence and Liouville theorems for V -harmonic maps from complete manifolds / Annals of Global Analysis and Geometry 42(4):565~584
B. Gidas / 1981 / Global and local behavior of positive solutions of nonlinear elliptic equations / Communications on Pure and Applied Mathematics 34(4):525~598
Zongming Guo / 2006 / Hausdorff Dimension of Ruptures for Solutions of a Semilinear Elliptic Equation with Singular Nonlinearity / manuscripta mathematica 120(2):193~209
Li Jiayu / 1991 / Gradient estimate for the heat kernel of a complete Riemannian manifold and its applications / Journal of Functional Analysis 97(2):293~310
Qun Chen / 2012 / Existence and Liouville theorems for V -harmonic maps from complete manifolds / Annals of Global Analysis and Geometry 42(4):565~584
B. Gidas / 1981 / Global and local behavior of positive solutions of nonlinear elliptic equations / Communications on Pure and Applied Mathematics 34(4):525~598
Zongming Guo / 2006 / Hausdorff Dimension of Ruptures for Solutions of a Semilinear Elliptic Equation with Singular Nonlinearity / manuscripta mathematica 120(2):193~209
Li Jiayu / 1991 / Gradient estimate for the heat kernel of a complete Riemannian manifold and its applications / Journal of Functional Analysis 97(2):293~310
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