Let $ \Omega$ be a $ C^1$ bounded domain in $ \mathbb{R}^3$ . Suppose that $ u:\Omega \rightarrow \mathbb{R}$ satisfies

$ -\Delta u \leq |u|^\gamma $ on $ \Omega$ , $ u=0$ on $ \partial \Omega$ .

Can we obtain that all the functions $ u$ enjoy some uniform bound from above?

I’m thinking about taking the equality case first and just consider the equation

$ -\Delta u = |u|^\gamma $ on $ \Omega$ , $ u=0$ on $ \partial \Omega$

and see if there exist some constant $ C$ such that for all the solutions to this equation, there holds

$ |u| \leq C$ , for all $ x \in \Omega$

But I’m not sure if it make sense at all…Could anyone shed some light on it? Any reference or partial answers are welcome. Thanks in advance!

Update: Sorry I forgot to mention the range for $ \gamma$ . It’s $ \gamma >3$ .