In the section \textbf{Laplace equation in half-plane} it says

"The problem $u_{yy}+u_{xx}=0$, $y>0$, $u(x,0)=g(x)$ [...] is not uniquely solvable". As an example the function $u(x,y)=y$ is given, which satisfies the Laplace equation. But it does obviously not satisfy $u(x,0)=g(x)$, so I don't see how we can conclude immediately that only bounded solutions are unique...

Can anybode help me with this?

Thanks!