A Harnack type inequality for the Yamabe equation in low dimensions (Q1890016)

Let \(B_1\subset\mathbb{R}^n\) be the unit ball centered at the origin, \(n\geq 3\), and \((a_{ij}(x))\) a smooth, \(n\times n\) symmetric positive definite matrix function on \(B_1\), satisfying \(\frac12| \xi| ^2\leq a_{ij}(x)\xi^i\xi^j\leq 2| \xi| ^2\) for any \((x,\xi)\in B_1\times\mathbb{R}^n\) and \(| | a_{ij}| | _{C^3(B_1)}\leq\bar{a}\) for some \(\bar{a}>0\). Let \(g\) be the Riemannian metric on \(B_1\) associated to \((a_{ij}(x))\) and consider the Yamabe equation \(-L_gu=u^{\frac{n+2}{n-2}}\), \(u>0\), on \(B_1\), where \(L_g\) is the conformal Laplacian of \(g\). For \(n=3,4\), the authors prove the existence of some positive constants \(\delta\) and \(C_0\), depending only on \(\bar{a}\), such that any smooth solution \(u\) of Yamabe equation satisfies the Harnack type inequality \(\displaystyle{\sup_{B(0,\varepsilon)}u\cdot \inf_{B(0,4\varepsilon)}u\leq C_0\varepsilon^{2-n}}\) for any \(\varepsilon\in (0,\delta)\), where \(B(0,\varepsilon)\) denotes the geodesic ball, with respect to \(g\), centered at \(0\) with radius \(\varepsilon\). For \(n=3,4\), a consequence is the estimate \(\int_{B_{1/2}}u^{2n/(n-2)}\leq C(\bar{a})\) for any smooth solution \(u\) of Yamabe equation. The main task in authors' proof, based on the method of moving planes, is to produce suitable auxiliary functions so that this method applies. The construction of the auxiliary functions for \(n=4\) is more delicate than that for \(n=3\), making use of coordinates with special properties. The authors argue by contradiction, and therefore their proof does not yield explicit constants \(\delta\) and \(C_0\) (see also, \textit{A. Li} and the first author [C. R., Math., Acad. Sci. Paris 336, No. 4, 319--324 (2003; Zbl 1113.35069)].