Does there exist a weak solution in $W^{1,2}_0(\Omega)$ to the following equation: $-\Delta u+bu=f$?

Question

Let $\Omega$ be a smooth bounded open subset in $\mathbb{R^3}$, $f$ and $b$ be in $L^2(\Omega)$ such that $b$ is non-negative on $\Omega$. Does there exist a weak solution in $W^{1,2}_0(\Omega)$ to the following equation: $-\Delta u+bu=f$?

Help me some hints to start. Furthermore, tell me some textbook about PDEs which contains many problems like this problem.

Thanks a lot.

Answer 1

Hint: is the bilinear form $$ a(u,v)=\int_\Omega(\nabla u\cdot\nabla v+b\,u\,v),\quad u,v\in W_0^{1,1}(\Omega), $$ coercive?

There are excelent books by L.C. Evans and H. Brèzis.