I have a big problem to solve this system:

$\Delta f−hf^2=0$

$p|\nabla f|^2+hf^3=0$

where $h$ and $p$ are constants (with $h \neq 0$ and $p \neq 0$, $p \neq -1$), $f$ is a scalar function defined on a 4-manifold ($f:M \rightarrow \mathbb{R}$) where $M$ is a 4-manifold not compact and where $\Delta f$ is the Laplacian of $f$ (trace of Hessian of $f$, with positive sign, not negative), and $\nabla f$ is the gradient of $f$ for the metric $g$ (where $g$ is the metric of $M$).

Should I find $f$ and $g$ excluding cases of flat metric $g$.

1. Are there solutions?

2. Can the metric $g$ admit a scalar curvature ($S$) equal to $-p(p+1)x$, i.e., $S=-p(p+1)x$, for some negative constant $p$ ?