My question is : 

> 
> Let $\mu=\mathbb{R}\setminus\left\{ 0\right\} $. Consider $$\left(\*\right)\begin{cases} -\triangle u+\mu u=f & U\\ u=0 & \partial U. 
> \end{cases} $$
>  Define 
>  $$
> \lambda\_{1}=\inf\_{u\in H\_{0}^{1}}\frac{\int\_{U}|Du|^{2}\ dx}{\int\_{U}u^{2}\ dx}
> $$ If $\mu>-\lambda\_{1}$, want to prove that $\left(\*\right)$ has a unique
>  weak solution $u\in H\_{0}^{1}\left(U\right)$ for each $f\in
> > L^{2}\left(U\right)$.
> 
> 
> 

I know the Lax-Milgram theorem and some existence theorem in Chapter
6, Evans.

If I let 
$$
B[u,v]=\int\_{U}-\triangle u\cdot v+\mu uv\ dx,
$$
then 
$$
B[u,v]=\int\_{U}Du\cdot Dv\ dx+\mu\int\_{U}uv\ dx.
$$
To show the assumption on Lax-Milgram theorem, clearly it satisfies
symmetry and uniform ellipticity. And readily, I proved that 
$$
|B[u,v]\leq\alpha\|u\|\_{H\_{0}^{1}}\|v\|\_{H\_{0}^{1}}.
$$
So, $B[u,u]\geq\beta\|u\|\_{H}^{2}$ is left to me.

For this, 
$$
B[u,u]=\int\_{U}|Du|^{2}+\mu\int\_{U}u^{2}>\int\_{U}|Du|^{2}-\lambda\_{1}\int\_{U}u^{2}\geq0.
$$

This one never gives me anyting.

How can I go further? Or, do I have to take other way?