I have a big problem to solve this system

 $\Delta f-hf^2=0$

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

where $h$ is a constant, $f$ is a 2-dimensional smooth function, $\Delta f$ is Laplacian of $f$ (i.e. $\Delta f=f_{xx}+f_{yy}$) and $\nabla f$ is the gradient of $f$.

ADD In first case $f$ is defined on $R^2$ and in second case $f$ is defined on surface $S$ ($f:S \rightarrow (0, \infty)$). is there a solution? Thank you for help

I have a big problem to solve this system  $\Delta f-hf^2=0$

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

where $h$ is a constant, $f$ is a 2-dimensional smooth function, $\Delta f$ is Laplacian of $f$ (i.e. $\Delta f=f_{xx}+f_{yy}$) and $\nabla f$ is the gradient of $f$.

ADD In first case $f$ is defined on $R^2$ and in second case $f$ is defined on surface $S$ ($f:S \rightarrow (0, \infty)$). is there a solution? Thank you for help

MODIFICATION after Igor Khavkine answer: and if the system is

$\Delta f-hf^2+cf=0$

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

(c is another constant)

I have a big problem to solve this system

$\Delta f-hf^2=0$

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

where $h$ is a constant, $f$ is a 2-dimensional smooth function, $\Delta f$ is Laplacian of $f$ (i.e. $\Delta f=f_{xx}+f_{yy}$) and $\nabla f$ is the gradient of $f$.

is there a solution? Thank you for help

I have a big problem to solve this system

$\Delta f-hf^2=0$

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

where $h$ is a constant, $f$ is a 2-dimensional smooth function, $\Delta f$ is Laplacian of $f$ (i.e. $\Delta f=f_{xx}+f_{yy}$) and $\nabla f$ is the gradient of $f$.

ADD In first case $f$ is defined on $R^2$ and in second case $f$ is defined on surface $S$ ($f:S \rightarrow (0, \infty)$). is there a solution? Thank you for help