 Zero-boundary harmonic function with pointwise vertical decay

The imaginary part of the entire function $e^{z^2}$ is a harmonic function. On the upper half-plane it vanishes at $y=0$ and tends to zero as $y\to \infty$ for every fixed $x$, but is nonzero and grows rapidly in the horizontal direction. Adding it to a Poisson integral preserves those boundary and vertical-decay conditions. A boundedness or suitable transform-growth condition is therefore needed to make the half-plane Dirichlet problem unique.