$Lu=\Delta u+hu$ is an elliptic operator with $h$ smooth on $\overline{\Omega}$. $u_{\alpha}$ is a family of solution of $Lu=0$ on a bounded domain $\Omega$.And we know $|\int_{\Omega}u_{\alpha}|\leq C.$Can we derive that $u_{\alpha}$ is uniformly bounded on $\Omega$?
This is a step in a paper I'm reading recently.The author says this is true but I cannot figure it out.Any help will be thanked.