I want to get the boundedness of (weak) solution $u$ to the following elliptic PDE with Robin boundary condition: \begin{equation} \left\{ \begin{aligned} - \Delta u + u & = f \qquad in~ \Omega, \\ \frac{\partial u}{\partial n} + \gamma u & = g \qquad on~\Gamma, \end{aligned} \right. \end{equation}
Assumptions:
$\Omega \subset \mathbb{R}^d~(d=2,3)$ is an open bounded set with Lipschitz continuous boundary $\Gamma$.
the Robin coefficient $\gamma \in L^2(\Gamma)$ is bounded, i.e, $0 < c_{min} \leq \gamma \leq c_{max} < \infty$
$g \in H^{1/2}(\Gamma)$, $f \in L^2(\Omega)$.
If the equation was $-\Delta u = f$, I might try proceeding with Maximum/Minimum Principle. But in this case I got stuck.
I tried estimating $u_{H^1(\Omega)}$, but then realized boundedness of $u_{H^1(\Omega)}$ does not impliy boundedness of $u$ itself.
Any hints will be helpful. Thank you.