Let $\Omega$ be a smooth bounded domain. I have the nonlinear equation $$-\Delta u + u + c_0u = f$$ where $c_0:\Omega \to \mathbb{R}$ is a function. I don't have a sign on $c_0$. The data $f \in H^{-1}$.
Is there any chance for existence of solutions to this equation in the usual Sobolev spaces? The equation is linear but I can't apply Lax-Milgram since the $c_0$ term may not yield coercivity of the bilinear form.