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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 $h \neq0$ and $p \neq0$ and $\neq -1$), $f$ is a scalar function defined on a 3-manifold ($f:M \rightarrow \mathbb{R}^3$, where $M$ is a 3-manifold not compact), $\Delta f$ is Laplacian of $f$ and $\nabla f$ is the gradient of $f$, for the metric $g$ (where $g$ is the metric of $M$).

Should I find $f$ and $g$..is it possible? excluding cases of flat metric $g$, are there solutions?

Professor Robert Bryant has found the solution here (where $M$ is a 2-manifold): Pde system problem

EDITED AFTER Thomas Richard's comment

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 $h \neq0$ and $p \neq0$ and $\neq -1$), $f$ is a scalar function defined on a 3-manifold ($f:M \rightarrow \mathbb{R}$, where $M$ is a 3-manifold not compact), $\Delta f$ is Laplacian of $f$ and $\nabla f$ is the gradient of $f$, for the metric $g$ (where $g$ is the metric of $M$).

Should I find $f$ and $g$..is it possible? excluding cases of flat metric $g$, are there solutions?

Professor Robert Bryant has found the solution here (where $M$ is a 2-manifold): Pde system problem

EDITED AFTER Thomas Richard's comment

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 $h \neq0$ and $p \neq0$ and $\neq -1$), $f$ is a scalar function defined on a 3-manifold ($f:M \rightarrow \mathbb{R}^3$, where $M$ is a 3-manifold not compact), $\Delta f$ is Laplacian of $f$ and $\nabla f$ is the gradient of $f$, for the metric $g$ (where $g$ is the metric of $M$).

Should I find $f$ and $g$..is it possible? excluding cases of flat metric, are there a solutions?

Professor Robert Bryant has found the solution here (where $M$ is a 2-manifold): Pde system problem

EDITED AFTER Thomas Richard's comment

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 $h \neq0$ and $p \neq0$ and $\neq -1$), $f$ is a scalar function defined on a 3-manifold ($f:M \rightarrow \mathbb{R}^3$, where $M$ is a 3-manifold not compact), $\Delta f$ is Laplacian of $f$ and $\nabla f$ is the gradient of $f$, for the metric $g$ (where $g$ is the metric of $M$).

Should I find $f$ and $g$..is it possible? excluding cases of flat metric $g$, are there solutions?

Professor Robert Bryant has found the solution here (where $M$ is a 2-manifold): Pde system problem

EDITED AFTER Thomas Richard's comment

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 $h \neq0$ and $p \neq0$ and $\neq -1$), $f$ is a scalar function defined on a 3-manifold ($f:M \rightarrow \mathbb{R}^3$, where $M$ is a 3-manifold not compact), $\Delta f$ is Laplacian of $f$ and $\nabla f$ is the gradient of $f$, for the metric $g$ (where $g$ is the metric of $M$).

Should I find $f$ and $g$..is it possible? is there a solution?

Professor Robert Bryant has found the solution here (where $M$ is a 2-manifold): Pde system problem

EDITED AFTER Thomas Richard's comment

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 $h \neq0$ and $p \neq0$ and $\neq -1$), $f$ is a scalar function defined on a 3-manifold ($f:M \rightarrow \mathbb{R}^3$, where $M$ is a 3-manifold not compact), $\Delta f$ is Laplacian of $f$ and $\nabla f$ is the gradient of $f$, for the metric $g$ (where $g$ is the metric of $M$).

Should I find $f$ and $g$..is it possible? excluding cases of flat metric, are there a solutions?

Professor Robert Bryant has found the solution here (where $M$ is a 2-manifold): Pde system problem

EDITED AFTER Thomas Richard's comment