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Register a new userGradient Estimates For $Delta u + a(x)ulog u + b(x)u = 0$ and its Parabolic Counterpart Under Integral Ricci Curvature Bounds
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In this paper, we consider a class of important nonlinear elliptic equations $$Delta u + a(x)ulog u + b(x)u = 0$$ on a collapsed complete Riemannian manifold and its parabolic counterpart under integral curvature conditions, where $a(x)$ and $b(x)$ are two $C^1$-smooth real functions. Some new local gradient estimates for positive solutions to these equations are derived by Mosers iteration provided that the integral Ricci curvature is small enough. Especially, some classical results are extended by our estimates and a few interesting corollaries are given. Furthermore, some global gradient estimates are also established under certain geometric conditions. Some estimates obtained in this paper play an important role in a recent paper by Y. Ma and B. Wang [17], which extended and improved the main results due to B. Wang [29] to the case of integral Ricci curvature bounds.
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In this paper, we consider the gradient estimates of the positive solutions to the following equation defined on a complete Riemannian manifold $(M, g)$ $$Delta u + au(log u)^{p}+bu=0,$$ where $a, bin mathbb{R}$ and $p$ is a rational number with $p=frac{k_1}{2k_2+1}geq2$ where $k_1$ and $k_2$ are positive integer numbers. we obtain the gradient bound of a positive solution to the equation which does not depend on the bounds of the solution and the Laplacian of the distance function on $(M, g)$. Our results can be viewed as a natural extension of Yaus estimates on positive harmonic function.
In this short note, we use a unified method to consider the gradient estimates of the positive solution to the following nonlinear elliptic equation $Delta u + au^{p+1}=0$ defined on a complete noncompact Riemannian manifold $(M, g)$ where $a > 0$ and $ p <frac{4}{n}$ or $a < 0$ and $p >0$ are two constants. For the case $a>0$, this improves considerably the previous known results except for the cases $dim(M)=4$ and supplements the results for the case $dim(M)leq 2$. For the case $a<0$ and $p>0$, we also improve considerably the previous related results. When the Ricci curvature of $(M,g)$ is nonnegative, we also obtain a Liouville-type theorem for the above equation.
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For Riemannian manifolds with a smooth measure $(M, g, e^{-f}dv_{g})$, we prove a generalized Myers compactness theorem when Bakry--Emery Ricci tensor is bounded from below and $f$ is bounded.
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