find a weak solution in an intersection of Sobolev spaces

Question

In using-lax-milgram-to-find-a-weak-solution-in-an-intersection-of-sobolev-spaces 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 and got solution by dear @Tomas (thanks to him) link

Answer 1

Minus the bi-Laplacian, i.e. your operator $-\Delta^2$, is negative semidefinite (indeed, negative definite with your boundary conditions) on $L^2(U)$ as soon as you can define $-\Delta^2$ weakly by applying Gauss-Green formulae (this is certainly possible under mild assumptions on the boundary of your domain).

Therefore, 1 is certainly outside the spectrum of your operator, i.e., $$ 1-(-\Delta^2) $$ has a bounded inverse: This is all you need. You will certainly have a weak solution in $H^2(U)$. If your boundary is nice enough that you can define your operator not merely in a weak sense, but on the natural domain $H^4(U)$, then for any $f\in L^2(U)$ your solution will be even in $H^4(U)$.