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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.

micha
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  • If $f\ge 0$ you can get existence: the equation $-\Delta u + \max(u,0)^2 = f$ is solvable, this solution is nonnegative, and solves the original equation. – daw Nov 23 '23 at 15:42
  • Can you give me any hints or guide me to any results how to solve your equation? – micha Nov 23 '23 at 15:48
  • $u\mapsto \max(0,u)^2$ is monotone – daw Nov 23 '23 at 17:37
  • @micha, you can use the same method to prove in https://math.stackexchange.com/questions/2118371/existence-and-uniqueness-of-delta-u-u3-f?noredirect=1&lq=1 – xpaul Dec 27 '23 at 14:32
  • @xpaul , I do not think it is possible. In the other thread they can derive a priori estimates involving a term $|u|^4$. This is not possible here since we get $u^3$ and we are not allowed to take absolute values $|u|^3$. – micha Jan 08 '24 at 12:39

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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.

micha
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