Gradient Estimates for Lichnerowicz-Type Equations

Abstract

In this paper, we first study carefully the positive solutions to

\mathrm{\Delta }u+{\lambda }_{1}ulnu+{\lambda }_2{u}^{\alpha +1}=0

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Delta u+\lambda _{1}u\ln u +\lambda _{2}u^{\alpha +1}=0$$\end{document}

defined on a complete non-compact Riemannian manifold (M, g) with

Ric(g)⩾-Kg

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Ric(g)\geqslant -Kg$$\end{document}

, which can be regarded as Lichnerowicz-type equations, and according to the different parameter values in the equation, seven cases are discussed to obtain the gradient estimates of positive solutions to these equations which do not depend on the bounds of the solutions and the Laplacian of the distance function on (M, g). For the case

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0<\alpha <\frac{2}{n}$$\end{document}

, this improves considerably the previous related results. Moreover, we also obtain the Liouville-type result for these equations when

Ric(g)⩾0

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Ric(g)\geqslant 0$$\end{document}

and establish the Harnack inequality as consequences.

Keywords

Gradient estimate / Ricci curvature / Lichnerowicz-type equation / Harnack inequality / Nonlinear elliptic equations / 35B51

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