In Section $9.2$ Theorem $5$ of Lawrence Evans' Partial Differential Equations, First Edition the author proves that for a large enough $\lambda$, the equation
$\begin{array}-\Delta u+b(\nabla u)+\lambda u=0\ &\mbox{ in } U\\ u=0&\mbox{on }\partial U\end{array}$
has a solution in $H_0^1(U)$.
On page 507, the author writes
$\int_UC(|\nabla u|+1)|u|dx\leq\frac{1}{2}\int_U|\nabla u|^2dx+C\int_U(|u|^2+1)dx\ \mbox{ for }u\in H_0^1(U).$
Here $C$ is the Lipschitz constant for the Lipschitz function $b$.
My problem is that I cannot show this no matter how much I try. How is the gradient term becoming independent of $C$? Could someone please help!