A note on scalar curvature type problems in dimension 4 (Q929547)

For a 4-dimensional compact Riemannian manifold \((M,g)\) let \(a\), \(f\) be positive \(C^\infty\)-functions. It is proved that the problem \(\Delta_g u+ a(x)u= f(x)u^3\) always admits a positive solution, up to a conformal deformation of \(g\). This leads to a geometric obstruction result for the prescribed scalar curvature \(S_g\) problem, which can be stated as follows: Let \(f\) be a nonzero, nonnegative function and let \(a={1\over 6} S_g\). If the above problem admits a positive solution for some \(h\) isometric to \(g\), then there exists no metric \(g_0\in [g]\) such that \(S_{g_0}\leq 0\) on \(M\).