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Any help will be appreciated thanks!

Consider $(\mathbb{R^n},g)$ to be a Riemannian manifold. For simplicity we can assume the manifold to be asymptotically euclidean outside a compact domain $\Omega$.

Let F be a smooth function with as much regularity imposed on it as one may need...

Consider solving for Biharmonic functions u satisfying: $ \triangle_g \triangle_g u = \triangle_g F $ with $ u \in H^2_0 (\Omega)$ there clearly exists a unique solution to this equation and the proof is a standard use of lax-milgram theorem.

Now let us ask the following question: Does there exist a global bi harmonic function u in this manifold such that $u|_{\partial\Omega} = \partial_{\nu}u|_{\partial\Omega}= 0$

i.e: $ \triangle_g \triangle_g u = \triangle_g F $ in all of $\mathbb{R^n}$ and furthermore $u|_{\partial\Omega} = \partial_{\nu}u|_{\partial\Omega}= 0$ ...

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