Nonexistence of subsolutions of a nonlinear elliptic equation on bounded domains in a Riemannian manifold
Given a Riemannian manifold \((M,g)\) of dimension \(n\geq 3\) and a smooth function \(f\) on \(M\), \(f\) is the scalar curvature of a metric \(\widetilde g\) conformal to \(g\) if and only if there is a smooth positive solution \(u\) of \[ - {{4(n-1)}\over{(n-2)}} \Delta_g u + S_g u = fu^{(n+2)/(n-2)} \quad\text{on}\quad M, \tag{\(*\)} \] where \(\Delta_g\) and \(S_g\) are the Laplace-Beltrami operator and scalar curvature of \((M,g)\) respectively. A large amount of research has been devoted to understanding the solvability of \((*)\) under various assumptions on \((M,g)\) and \(f\). Here the author considers a class of equations including \((*)\) on certain types of open Riemannian manifolds and proves some nonexistence results in the case that \(f\) is nonpositive and satisfies a suitable growth assumption near \(\partial M\). Of particular interest is the fact that there is no assumption on the dimension or regularity of \(\partial M\). The main theorems generalize and include a number of earlier results as special cases.
scientific article; zbMATH DE number 1251457