Gradient estimates and Liouville-type theorems for a weighted nonlinear elliptic equation

We consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a smooth metric measure space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(M, g,e^{-f}\,dv)$\end{document}(M,g,e−fdv): \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta_{f} u+au\log u+bu=0, $$\end{document}Δfu+aulogu+bu=0, where a, b are two real constants. When the ∞-Bakry–Émery Ricci curvature is bounded from below, we obtain a global gradient estimate which is not dependent on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$|\nabla f|$\end{document}|∇f|. In particular, we find that any bounded positive solution of the above equation must be constant under some suitable assumptions.


Introduction
Let (M, g) be an n-dimensional complete Riemannian manifold and f be a smooth function defined on M. Then the triple (M, g, e -f dv) is called a smooth metric measure space, where dv denotes the volume element of the metric g and e -f dv is called the weighted measure. On the smooth metric measure space (M, g, e -f dv), the m-Bakry-Émery Ricci curvature (see [1][2][3]) is defined by where m ≥ n is a constant, and m = n if and only if f is a constant. We define Then Ric f can be seen as the ∞-dimensional Bakry-Émery Ricci curvature. However, there are many differences between the m-Bakry-Émery Ricci curvature and the ∞-Bakry-Émery Ricci curvature. For example, there exist complete noncompact Riemannian manifolds which satisfy Ric f = λg for some positive constant λ (which is called the shrinking gradient Ricci soliton), but not for Ric m f = λg. We recall that the f -Laplacian f on (M, g, e -f dv) is defined by Since we have the Bochner formula with respect to f -Laplacian: which is similar to the Bochner formula associated with the Laplacian, many results with respect to the Laplacian have been generalized to those of the f -Laplacian under the mdimensional Bakry-Émery Ricci curvature. For example, see [4][5][6][7] and the references therein. But for elliptic gradient estimates for f -Laplacian under the ∞-Bakry-Émery Ricci curvature, in order to using the weighted comparison theorem, the assumption |∇f | ≤ θ is forced commonly.
In this paper, under the assumption that the ∞-Bakry-Émery Ricci curvature is bounded from below, we consider the following nonlinear elliptic equation: where C is a positive constant which depends on the dimension n, β = max {x|d(x,p)=1} f r(x) and Letting R → ∞ in (1.4), we obtain the following global estimates on complete noncompact Riemannian manifolds: (1.7) Using the ideas of the proof of Theorem 1.1, by choosingh = log u a gap develops between the constants, and we also establish the following.
Then on B p (R) with R > 1, the following inequality holds: Letting R → ∞ in (1.8), we obtain the following global estimates on complete noncompact Riemannian manifolds: can be found in [9][10][11]. Moreover, Qian in [10] used a different method to derive similar estimates to (1.12) with constant f . On the other hand, if we assume Ric f ≥ -(n -1)K and |∇f | ≤ θ , then from (1.1), we obtain Hence, Theorem 1.5 in [11] follows from Theorem 1.1 of [11] immediately. However, our estimates in this paper are not dependent on |∇f |.

Proof of results
We firstly give the following lemma which plays an important role in the proof of main results. Since 0 <ũ ≤ 1, we have log h ≤ 0 and which implies Thus, under the assumption Ric f ≥ -(n -1)K , one has On the contrary, if ∇f ∇hah log h -b h ≥ δ |∇h| 2 h at the point p, then from (2.8), we can deduce a +b + (n -1)K + aL |∇h| 2 (2.10) as long as (2.1) holds.
Therefore, in these two cases the estimate (2.2) holds, which finishes the proof of the Lemma 2.1.

Proof of Theorem 1.3
We defineh = log u. Then we have Following the proof of Theorem 1.1 line by line, we obtain on B p (R) with R > 1, |∇h| 2 ≤ C 1 (n, δ, β) R + C 2 (n, δ) max a + (n -1)K, 0 , (2.29) where δ is taken to zero in the second assumption. We completed the proof of Theorem 1.3.