In 
http://math.stackexchange.com/questions/354032/using-lax-milgram-to-find-a-weak-solution-in-an-intersection-of-sobolev-spaces/544138#544138
the weak solution for 
$$
-\Delta^2 u = f \in L^2(U)\\ \\
u|_{\partial U}=\Delta u|_{\partial U} = 0
$$
was discussed, I have a question about the week solution of 
$$
\Delta^2 u + u = f \in L^2(U)\\ \\
u|_{\partial U}=\Delta u|_{\partial U} = 0
$$
I think I should use coupled elliptic PDE theory, Any hint or suggestion is helpful for me.

In advanced thanks from anyone who tries to help me.
I also asked this question in 
http://math.stackexchange.com/questions/354032/using-lax-milgram-to-find-a-weak-solution-in-an-intersection-of-sobolev-spaces/544138#544138