Uniqueness for the Brezis-Nirenberg problem on compact Einstein manifolds. (English) Zbl 1180.35232

Huang, Guangyue; Chen, Wenyi

Osaka J. Math. 45, No. 3, 609-614 (2008).

Let \((M^n,g_0)\) be the compact Einstein manifold with positive scalar curvature \(R_0\) and \(n\geq 3\). In the present study the authors consider the following nonlinear elliptic equation \[ \Delta_0u-\lambda n+f(u)u^{\frac{n+2}{n-2}}=0,\text{ on }M^n\quad u>0,\text{ on } M^n\tag{1} \] where \(\Delta_0\) is the Laplace-Beltrami operator on \(M^n\) related to \(g_0\). The authors prove that for \(0<\lambda<(n-2) R_0/4(n-1)\) and \(f(n)\leq 0\), and at least one of two inequalities is strict, the only positive solution of (1) is constant.

Reviewer: Messoud A. Efendiev (Berlin)

MSC:

35J60

Nonlinear elliptic equations

Keywords:

compact Einstein manifolds; positive solution; curvature; semi-linear elliptic equation

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