Dirichlet problem for upper half plane

The Dirichlet problem I read is as follows:

If $f$ is an integrable function, find a function $u$ such that for $x \in \mathbb{R}, y>0$ \begin{align}u_{xx} + u_{yy} & =0 \\\lim_{y \to 0^+} u(x,y) &= f(x) \text{ almost everywhere}\end{align}

Does the method listed in this Find the solution of the Dirichlet problem in the half-plane y>0.also work if $u$ and $u_x$ are not required to vanish as $|x| \to \infty$ and $u$ is bounded? Since if we discard these conditions we can not exclude the case $\lambda>0$.