Gradient estimates for the elliptic and parabolic Lichnerowicz equations on compact manifolds. (English) Zbl 1223.58013
Summary: Let \((M, g)\) be a smooth compact Riemannian manifold of dimension \(n \geq 3\). Denote by \({\Delta_g=-\mathrm {div}_g\nabla}\) the Laplace-Beltrami operator.
We establish some local gradient estimates for the positive solutions of the Lichnerowicz equation
\[ \Delta_g u(x)+h(x)u(x)=A(x)u^p(x)+\frac{B(x)}{u^q(x)} \]
on \((M, g)\). Here, \(p, q \geq 0\), \(A\), \(B\) and \(h\) are smooth functions on \((M, g)\). We also derive the Harnack differential inequality for the positive solutions of
\[ u_t(x,t)+\Delta_gu(x,t)+h(x)u(x,t)=A(x)u^p(x,t)+\frac{B(x)}{u^q(x,t)} \]
on \((M, g)\) with initial data \(u(x, 0) > 0\).
We establish some local gradient estimates for the positive solutions of the Lichnerowicz equation
\[ \Delta_g u(x)+h(x)u(x)=A(x)u^p(x)+\frac{B(x)}{u^q(x)} \]
on \((M, g)\). Here, \(p, q \geq 0\), \(A\), \(B\) and \(h\) are smooth functions on \((M, g)\). We also derive the Harnack differential inequality for the positive solutions of
\[ u_t(x,t)+\Delta_gu(x,t)+h(x)u(x,t)=A(x)u^p(x,t)+\frac{B(x)}{u^q(x,t)} \]
on \((M, g)\) with initial data \(u(x, 0) > 0\).
MSC:
| 58J05 | Elliptic equations on manifolds, general theory |
| 35B09 | Positive solutions to PDEs |
| 35L05 | Wave equation |
| 58J35 | Heat and other parabolic equation methods for PDEs on manifolds |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 35K05 | Heat equation |
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