My question is very similar to the one given here: Existence and uniqueness of $-\Delta u + u^3 =f$
If we consider the equation $-\Delta u+u^2=f$ in 3 dimensions with $f\in L^2(\Omega)$ is there anyhting we can say about possible solutions? The theory of monotone operators, the direct method or any other technique I have seen so far does not cover this case.
It is not possible to solve the equation in the above generality. For example take $f=-1$ and Neumann boundary conditions of the form $\partial_n u=0$. Then we can test the above equation by $f$ and get after integration by parts $$\int_\Omega -u^2=\int_\Omega 1>0$$ which is a contradiction. This suggests that it is important to set further assumptions if one hopes to solve an equation of the above type.