I have a question related to the Positivity Preserving Principle (PPP) for $ \Delta^2$ and related topics. Recall if $u$ solves $$\Delta^2 u = f(x) \mbox{ in } \Omega, \quad u=\partial_\nu u =0 \mbox{ on } \partial \Omega$$ we say it has (PPP) provided $ f \ge 0 $ in $ \Omega$ implies $ u \ge 0$ in $ \Omega$.

For $ \Omega$ a the unit ball this is true and for domains close to the unit ball (in some sense) its true (I think). For general smooth domains it can be false.

QUESTION. Take $ p>1$ (restrict if you like). I am interested if the only classical solution of $ \Delta^2 u + | \nabla u|^p=0 $ in $ \Omega$ with $u=\partial_\nu u=0$ on $ \partial \Omega$ is $u=0$.

Of course by scaling you can put a term $ \lambda \neq 0$ in front of the term $| \nabla u|^p$ if one desires. Put whatever assumptions on $ \Omega$ that one desired also. Sorry if this is a trivial problem.



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