I couldn't find any example online where the Laplace equation was solved by a Laplace Transform at some point in the solution but perhaps @fresh_42 or @Mark44 know of one.
the Laplace transform of the partial derivative is ##L[\frac{\partial^2U}{\partial x^2}] = \frac{d^2u}{dx^2}##. This means that the Laplace transform is not useful in solving the Laplace equation, but it can be used to solve the heat equation, wave equation, and basically any 2D PDE for U(x,t) where one partial derivative is with respect to time and the other with respect to the spatial coordinate.
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