Gradient estimates for a class of elliptic equations with logarithmic terms

We obtain the gradient estimates of the positive solutions to a nonlinear elliptic equation on an n -dimensional complete Riemannian manifold ( M , g )


Introduction
Let (M, g) be an n-dimensional complete Riemannian manifold.Recently, many authors studied the following elliptic differential equation u + au ln u = 0 on M, (1.1) 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 u + au(ln u) p + bu ln u = 0 on M, (1.3) where a = 0, b are two constants and p = k 1 2k 2 +1 ≥ 2, here k 1 and k 2 are two positive integers.In the case a ≡ 0 and b ≡ 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 ≡ 0 and b < 0, Ma [11] studied the gradient estimates of the positive solutions to the above elliptic equation for dim(M) ≥ 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 ≡ 0 and b ∈ R. Recently, Peng, Wang, and Wei [12] considered the following equation u + au(ln u) p + bu = 0, where a, b ∈ R and p = k 1 2k 2 +1 ≥ 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 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 Ric(g) ≥ -Kg, where the constant K := K(2R) ≥ 0 in the geodesic ball where k 1 and k 2 are two integers, and 1 < λ < 2. Let u(x) be a smooth positive solution to (1.3).Then we have, in where (ii) when a < 0, where 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.u ≤ e max{1,S}R inf u. Here, , where H 1 , Y 1 and Y 2 are constants in Theorem 1.1 with λ = 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
where k and k 2 are two integers, we see Here, where H 1 , Y 1 and Y 2 are constants in Theorem 1.1 with λ = 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.

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 Ric(g) ≥ -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 Proof In a normal coordinate at point O, we have It follows that Using the Bochner-Weitzenböck formula and Ric(g) ≥ -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.
where C 1 and C 2 are two positive constants independent of (M, g).

Proof of Theorem 1.1
In this section, we will prove Theorem 1.1.

It follows that
Case 2: If a > 0, w p-1 < 0 and w < 0, then from (3.3), we have where By the definition of Y 1 , we get By the Young inequality again, we find and 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 where    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.

Proof of Corollaries
In this section, we will prove Corollary 1.2 (the Harnack inequality) and Corollary 1.