A new local gradient estimate for a nonlinear equation under integral curvature condition on manifolds (Q785751)

In this article, the authors consider the nonlinear equation \[ \Delta u + au \log u + bu = 0 \] on a Riemannian manifold \(M^n\), where \(a \ge 0, b\) are two real constants. Replacing \(u\) by \(e^{-b/a}u\), the equation reduces to \[ \tag{\text{1-1}}\Delta u + au \log u = 0. \] For each \(x\in M\), the authors assume \(\varrho(x)\) denotes the smallest eigenvalue for the Ricci tensor \(\mbox{Ric}: T_x M\to T_x M\) and \(\mbox{Ric}^K_-(x) =((n-1)K -\varrho(x)_+= \max\{0,(n-1)K -\varrho(x)\}\), the amount of Ricci curvature lying below \((n-1)/K\). Further \[ \|\mbox{Ric}^K_-\|_q= \sup_{x\in M} \Bigl( \int_{B_0(R)} (\mbox{Ric}^K_-)^q\, d \mbox{vol} \Bigr)^{1/q} \] and \[ k(x,q,R)= R^2 \Bigl( \fint_{B_0(R)} (\mbox{Ric}^K_-)^q \,d \mbox{vol} \Bigr)^{1/q}\,, \quad k(q, R)= \sup_{x\in R} k(x, q, R)\,, \] where \(\fint_{B_0(R)} \) means the average integral. As the main result the authors determine a local gradient estimate for positive solutions to (1-1). Theorem 1.1. Let \((M,g)\) be a complete Riemannian manifold. Suppose that \(u\) is a positive solution to (1-1) on the ball \(B_0(R)\subset M\). For \(q > n/2,\) \( K > 0\) and \(R\le 1\), if there exists a large enough constant \(b\) such that \(k(q, 1) \le 1/(b+1)^2\), then \[ \frac{|\nabla u|}{u}\le \frac{C}{R}\quad on \; B_0\Bigr(\frac{R}{2}\Bigl), \] where \(C\) is a positive constant.