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I know that the solutions to the Laplace equation $\triangle u=0$ are unique but in some cases one can find that there is more than 1 solution. For example $u=0$ and ln$|\textbf{x}|$ are both solutions to $\triangle u=0, u=0$ on $|\textbf{x}|=1$. How does this not contradict uniqueness?

dan
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    Notice that $\ln\lvert x\rvert$ is not defined on the disk, but rather on the disk minus its center. The domain of definition comes heavily into play in existence and uniqueness of solutions to PDEs. –  Aug 24 '21 at 22:02
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    Also boundary conditions are important for what the unique solution is. In your example you have not given any boundary conditions at $|\mathbf{x}|=0.$ Your two solutions have very different values there. – md2perpe Aug 24 '21 at 22:09

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