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Gradient estimates for a class of elliptic equations with logarithmic terms
Boundary Value Problems volume 2024, Article number: 40 (2024)
Abstract
We obtain the gradient estimates of the positive solutions to a nonlinear elliptic equation on an n-dimensional complete Riemannian manifold \((M, g)\)
where \(a\ne 0\), b are two constants and \(p=\frac{k_{1}}{2k_{2}+1}\ge 2\), here \(k_{1}\) and \(k_{2}\) are two positive integers. The gradient bound is independent of the bounds of the solution and the Laplacian of the distance function. As the applications of the estimates, we show the Harnack inequality and the upper bound of the solution.
1 Introduction
Let (M, g) be an n-dimensional complete Riemannian manifold. Recently, many authors studied the following elliptic differential equation
where a is a constant. This equation is closely related to the logarithmic Sobolev inequality [3, 6, 16]. It is also involved in the gradient Ricci solution [11, 13, 17] devoted to understanding the Ricci flow introduced by Hamilton [8].
In [1], Abolarinawa considered the following equation
on a complete smooth metric measure manifold with weight \(e^{-f}\) and Bakry-Emery Ricci tensor bounded from below, where a and α are constants. He obtained the local gradient estimates dependent on the bound of solutions. The importance of gradient estimates cannot be overemphasized in geometric analysis and mathematical physics. For instance, they can be used to find the Hölder continuity of solutions and estimate on the eigenvalues; see [4, 5, 14, 15] and references therein. In particular, Gui, Jian, and Ju [7] obtained the local gradient estimate and Liouville-type theorem of translating solutions to mean curvature flow.
In this paper, we study the local gradient estimate of the positive solution to the following more general nonlinear elliptic equation
where \(a\neq 0\), b are two constants and \(p=\frac{k_{1}}{2k_{2}+1}\geq 2\), here \(k_{1}\) and \(k_{2}\) are two positive integers.
In the case \(a\equiv 0\) and \(b\equiv 0\), (1.3) is the Laplace equation. The corresponding gradient estimate was established by Yau [18]. Later, Li and Yau [10] obtained the well-known Li-Yau estimate for the Schrödinger equation and derived a Harnack inequality. In the case \(a\equiv 0\) and \(b<0\), Ma [11] studied the gradient estimates of the positive solutions to the above elliptic equation for \(\operatorname{dim}(M) \geq 3\). Then, Yang [17] improved the estimate of [11] and extended it to the case \(b>0\), and M is of any dimension. Chen and Chen [2] also extended the estimate of [11] to the case \(b>0\). Later, Huang and Ma [9], Qian [13], Zhu and Li [19] also studied the gradient estimates of the positive solutions to the above elliptic or the corresponding parabolic equation in the case \(a\equiv 0\) and \(b\in \mathbb{R}\). Recently, Peng, Wang, and Wei [12] considered the following equation
where \(a,b\in \mathbb{R}\) and \(p=\frac{k_{1}}{2k_{2}+1}\geq 2\), here \(k_{1}\) and \(k_{2}\) are positive integers. They obtained the local gradient estimates and derived a Harnack inequality.
Throughout the paper, we use the notation \(\mathrm{Ric}(g)\) to denote the Ricci curvature of \((M,g)\). Now we state the local gradient estimates independent of the bounds of the solution and the Laplacian of the distance function.
Theorem 1.1
(Local gradient estimate) Let \((M,g)\) be an n-dimensional complete Riemannian manifold with \(\mathrm{Ric}(g)\ge -Kg\), where the constant \(K:=K(2R)\geq 0\) in the geodesic ball \(B_{2R}(O)\). Here, O is a point in M. Suppose \(a,b \in \mathbb{R}\), \(a\neq 0\), \(p=\frac{k_{1}}{2k_{2}+1}\ge 2\) where \(k_{1}\) and \(k_{2}\) are two integers, and \(1<\lambda <2\). Let \(u(x)\) be a smooth positive solution to (1.3). Then we have, in \(B_{R}(O)\),
(i) when \(a>0\),
where
with
(ii) when \(a<0\),
where
with
Here,
with \(C_{1}\) and \(C_{2}\) are two uniform positive constants,
and
As a consequence of Theorem 1.1, we have the following Harnack inequality.
Corollary 1.2
(Harnack inequality) Assume that the same conditions in Theorem 1.1hold. Then, for \(a>0\), \(b=0\), \(\lambda =\frac{3}{2}\), \(p=\frac{2k}{2k_{2}+1}\ge 2\) where k and \(k_{2}\) are two integers, we have
Here,
where \(H_{1}\), \(Y_{1}\) and \(Y_{2}\) are constants in Theorem 1.1with \(\lambda =\frac{3}{2}\).
As another application of Theorem 1.1, we show the upper bound of solutions, which is analogous to the result obtained by Qian [13].
Corollary 1.3
Assume that the same conditions in Theorem 1.1hold. Then, for \(a>0\), \(b\neq 0\), \(\lambda =\frac{3}{2}\), \(p=\frac{2k}{2k_{2}+1}\ge 2\) where k and \(k_{2}\) are two integers, we see
Here,
where \(H_{1}\), \(Y_{1}\) and \(Y_{2}\) are constants in Theorem 1.1with \(\lambda =\frac{3}{2}\).
The structure of this paper is as follows: In Sect. 2, we give some lemmas, which will be used in the following section. Section 3 is a proof of Theorem 1.1. The last section is devoted to the proof of Corollary 1.2 (a Harnack inequality) and Corollary 1.3.
2 Preliminaries
In this section, we first construct an auxiliary function and establish a differential inequality. Then, a lemma on cut-off functions is introduced. Suppose that an n-dimensional complete Riemannian manifold (M, g) satisfies \(\mathrm{Ric}(g)\geq -Kg\) in a geodesic ball \(B_{2R}(O)\), where \(K=K(2R)\) is a nonnegative constant, and O is a fixed point on M.
Lemma 2.1
Assume that \(u(x)\) is a smooth positive solution to (1.3) in a geodesic ball \(B_{2R}(O)\). Setting \(w=\ln u\) and
where \(1<\lambda <2\) and \(b\in \mathbb{R}\), we obtain
Proof
In a normal coordinate at point O, we have
It follows that
Using the Bochner-Weitzenböck formula and \(\mathrm{Ric}(g)\ge -Kg\), we get
By the definition of G and the Cauchy-Schwarz inequality, we see
From (2.1) and (2.2), we obtain this lemma. □
Lemma 2.2
([12], Lemma 2.2) Let ϕ be a cut-off function, that is, \(\phi (x)|_{B_{R}(O)}=1\), \(\phi (x)|_{M\setminus B_{2R}(O)}=0\). Then, ϕ satisfies
and
where \(C_{1}\) and \(C_{2}\) are two positive constants independent of (M, g).
3 Proof of Theorem 1.1
In this section, we will prove Theorem 1.1.
Proof of Theorem 1.1
Let \(x_{0}\in B_{2R} ( O )\) such that \(\phi G(x_{0} )=\sup_{B_{2R} ( O ) } (\phi G) \). If \(\phi G(x_{0})< 0\), then we finish the proof. Hence, we may assume that \(\phi G(x_{0} )> 0\). Note that \(x_{0}\notin \partial B_{2R}(O)\). It follows that \(\nabla (\phi G) ( x_{0} ) =0\), \(\Delta (\phi G) ( x_{0} ) \le 0\). By Lemma 2.2, we see that
where \(A=\frac{(n-1)(1+\sqrt{K}R )C_{1}^{2}+C_{2}+2C_{1} ^{2}}{R^{2}}\), and
By Lemma 2.1, we have
If \(G\le 1\), then by \(0\le \phi \le 1\), we have \(\phi G\le 1\). Thus, we may assume that \(G\ge 1\). For the clarity, we consider the following six cases to prove Theorem 1.1.
Case 1: \(a>0\), \(w^{p-1} > 0\) and \(w>0\); Case 2: \(a>0\), \(w^{p-1}<0\) and \(w<0\);
Case 3: \(a>0\), \(w^{p-1}>0\) and \(w<0\); Case 4: \(a<0\), \(w^{p-1}>0\) and \(w>0\);
Case 5: \(a<0\), \(w^{p-1}>0\) and \(w<0\); Case 6: \(a<0\), \(w^{p-1}<0\) and \(w<0\);
Now we discuss the above six possibilities on a case-by-case basis.
Case 1: If \(a>0\), \(w^{p-1} > 0\) and \(w>0\), then by the Young inequality, we have
and
Combining (2.3) and (3.3), we get
From (3.1), (3.2) and (3.4), we see
Case 1.1: If \(w\in [H_{1},+\infty )\), where \(H_{1}=\max \{ L_{1} ,V_{1} \}>0 \) with
then we have
By the definition of \(H_{1}\), we see
From (3.5), (3.6), (3.7), and (3.8), we obtain
By the Young inequality again, we have
and
Combining (3.9), (3.10), and (3.11), we get
Case 1.2: If \(w\in (0,H_{1} )\), then we see
and
Employing (3.5), (3.13), (3.14), (3.15) and \(G\ge 1\), we get
It follows that
Case 2: If \(a>0\), \(w^{p-1}<0\) and \(w<0\), then from (3.3), we have
By the Cauchy inequality, we see
Using (3.1), (3.17), and (3.18), we find
Case 2.1: If \(w\in (-\infty ,Y_{1} ]\), where
with
then we have
and
By the definition of \(Y_{1}\), we get
Combining (3.19), (3.20), (3.21), (3.22), and (3.23), we have
By the Young inequality again, we find
It follows from (3.24) and (3.25) that
Case 2.2: If \(w\in (Y_{1} ,0 )\), then we get
and
Employing (3.19), (3.27), (3.28), (3.29), and \(G\ge 1\), we obtain
Case 3: If \(a>0\), \(w^{p-1}>0\) and \(w<0\), then by the Young inequality, we see
Combining (3.1), (3.17), (3.18), and the above inequality, we get
Case 3.1: If \(w\in (-\infty ,Y_{2} ]\), where \(Y_{2} =\min \{J_{1} ,V_{2} \}\) with
then we have
and
From (3.31), (3.32), (3.33), and (3.34), we find
By the Young inequality again, we see
If \(-\frac{2(\lambda -1) G}{n} +2K\lambda +3\lambda \vert b \vert ( p-2 ) -\frac{2\lambda \vert b \vert p(p-1)}{w}<0\), then from (3.35), we have
If \(-\frac{2(\lambda -1) G}{n} +2K\lambda +3\lambda \vert b \vert ( p-2 ) -\frac{2\lambda \vert b \vert p(p-1)}{w} \ge 0\), then we get
It follows from (3.36) and (3.37) that
Case 3.2: If \(w\in (Y_{2},0)\), then we find
and
Employing (3.31), (3.39), (3.40), (3.41), and \(G\ge 1\), we have
From (3.12), (3.16), (3.26), (3.30), (3.38), and (3.42), we obtain the upper bound of Ï•G in the case \(a>0\).
We are able to get the estimates for Cases 4–6 along a similar line to Cases 1–3. Thus, we only state the results of Cases 4–6.
Case 4: If \(a<0\), \(w^{p-1}>0\) and \(w>0\), then we have
Case 5: If \(a<0\), \(w^{p-1}>0\) and \(w<0\), then we see
Case 6: If \(a<0\), \(w^{p-1}<0\) and \(w<0\), we get
From (3.43), (3.44), and (3.45), we see that the upper bound of ϕG in the case \(a<0\). □
4 Proof of Corollaries
In this section, we will prove Corollary 1.2 (the Harnack inequality) and Corollary 1.3.
Proof of Corollary 1.2
When \(a>0\), \(b=0\) and \(p=\frac{2k}{2k_{2}+1}\ge 2\), where k and \(k_{2}\) are two integers, setting \(\lambda =\frac{3}{2}\), then
Choose y, z in \(B_{\frac{R}{2} } ( O ) \) such that \(u ( y ) =\sup_{B_{\frac{R}{2} } ( O )}u ( x )\), \(u ( z ) =\inf_{B_{\frac{R}{2} } ( O )}u ( x ) \). Let \(\gamma ( t ) \in [ 0,l ]\) be a shortest curve connecting y and z with \(\gamma ( 0 ) =y\), \(\gamma ( l ) =z\). By the triangle inequality, we have \(\gamma \in B_{R} ( O ) \) and \(l\le R\). It follows from Theorem 1.1 that
where
Therefore, we obtain
 □
Proof of Corollary 1.3
When \(a>0\), \(b\neq 0\) and \(p=\frac{2k}{2k_{2}+1}\ge 2\), where k and \(k_{2}\) are two integers, setting \(\lambda =\frac{3}{2}\), then
and
It follows from Theorem 1.1 that
that is,
Here, Ŝ is the constant \(T_{1}\) in Theorem 1.1 with \(\lambda =\frac{3}{2}\), i.e.,
 □
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This work is supported by National Natural Science Foundation of China (12371209) and Beijing Natural Science Foundation (1232004).
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Gao, Z., Guo, Q. Gradient estimates for a class of elliptic equations with logarithmic terms. Bound Value Probl 2024, 40 (2024). https://doi.org/10.1186/s13661-024-01845-3
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DOI: https://doi.org/10.1186/s13661-024-01845-3