Let $\beta:\mathbb{R}\to\mathbb{R}$ be a Lipschitz function such that $0<a\leq \beta^\prime\leq b$, for some constants $a,b$. Give the weak formulation of the problem \begin{equation} \left\{\begin{array}{cl}-\Delta u=f&in\ \Omega\\\partial_\nu u+\beta(u)=0&on\ \partial\Omega,\end{array}\right. \end{equation} where $\Omega\subset\mathbb{R}^n$ is a bounded domain and $f\in L^2(\Omega)$. Show that there exists a weak solution that it is unique.
My guess is that you have to use Lax-Milgram in order to prove existence and uniqueness of solutions, but I don't know how to deal with the nonlinear term. The weak formulation of the problem reads \begin{equation} a(u,v)=\varphi(v)\ \forall v\in H^1(\Omega), \end{equation} where \begin{equation} \begin{array}{l} a(u,v)=\int_\Omega\nabla u\nabla v+\int_{\partial\Omega}\beta(u)v,\\ \varphi(v)=\int_\Omega fv. \end{array} \end{equation} $\varphi$ is clearly continuous, but how can I prove continuity and coercivity of the bilinear form $a(\cdot,\cdot)$?