Solution of Poisson's equation with Neumann boundary condition

Question

Consider the Poisson's equation with Neumann boundary condition \begin{cases}-\Delta u= f, &\text{ on } \Omega\\ \nabla u \cdot n = g &\text{ on } \partial \Omega\\ \end{cases} If $g=0$, we can apply Lax-Milgram theorem to show the existence of a unique solution in the space $$V=\{u\in H^1(\Omega):\int_{\Omega}u=0\}.$$ Now, consider for $g\neq0$, with compatibility condition $\int_{\Omega}f+\int_{\partial\Omega}g=0$. The weak formulation in $H^1(\Omega)$ is: Find $u\in H^1(\Omega)$ such that $$\int_{\Omega}\nabla u\cdot\nabla v=\int_{\Omega}fv+\int_{\partial\Omega}gv, \forall v\in H^1(\Omega).$$ I am trying to use Fredholm's alternative: Let us define $$B(u,v)=\int_{\Omega}\nabla u\cdot\nabla v+\int_{\Omega}uv.$$ Then by Lax-Milgram theorem there exists a unique solution $u\in H^1(\Omega)$ for $f\in L^2(\Omega), g\in L^2(\partial \Omega)$ to $$B(u,v)=\int_{\Omega}fv+\int_{\partial\Omega}gv, \forall v\in H^1(\Omega).$$ Now I do not know how to proceed for Fredholm's alternative.