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Let $D$ be the unit disc in $\mathbb R^2$ centered at the origin. Let $w \in C^{\infty}_c(D)$ satisfy $$ (1-r^2)^2\Delta w +w =0.$$ Prove that $w \equiv 0$.

Ali
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    You can expand $w(r,\varphi) = \sum w_n(r) e^{in\varphi}$, and then the $w_n(r)$ satisfy ODEs and $w_n=0$ near $r=1$, so $w_n\equiv 0$. – Christian Remling May 20 '21 at 20:05
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    For what it's worth, Ali, if you are specifically looking for a 'unique continuation'-type argument, I believe this is addressed in this answer in a more general setting. – Leo Moos May 22 '21 at 19:47

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