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 want 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, that I have no idea about.

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