A saddle point problem for Poisson's equation

Question

I want to prove the well-posedness of the following saddle point problem: Given $f\in L^{2}(\Omega)$ , $\alpha \in \mathbb{R}$. Consider the problem of finding $(u,\lambda)\in H_{0}^{1}\times \mathbb{R}$ such that $$ \begin{cases} -\Delta u(x)+\lambda=f(x) & x\in\Omega\\ \\ \displaystyle\int\limits_{\Omega} u(x)\,\mathrm{d}x=\alpha \end{cases} $$ My problem is that this equation is similar to Poisson equation but I don't know what to do with $\lambda$: especially I have problem in writing the weak formulation and proving its well-posedness.