Let $n \geq 1, \lambda >0, f \in L^2(\mathbb{R}^n)$ and $b: \mathbb{R}^n \to \mathbb{R}^n$ a vector field satisfying $\mbox{div}(b) \in L^\infty(\mathbb{R}^n)$. What would be the most natural approach to solve the following equation: $\lambda u -\frac{1}{2} \Delta u + b(x)\nabla u = f \quad \mbox{in} \quad \mathbb{R}^n$ ?
Obvisously, when $\lambda$ is larger than $\| \mbox{div}(b) \|_{\infty}$ then Lax-Milgram theory can be applied to get a solution. But what can we say for $\lambda$ smaller than $\| \mbox{div}(b) \|_{\infty}$?
