For a concrete example, the dipole potential $$ \dfrac{x_1}{(x_1^2 + x_2^2 + x_3^2)^{3/2}}$$
is harmonic and $L^2$ on $a < x_1 < b$ if $0 < a < b$.  Higher multipole potentials such as 
$$ \dfrac{2 x_1^2 - x_2^2 - x_3^2}{(x_1^2 + x_2^2 + x_3^2)^{5/2}}$$ 
are harmonic and $L^2$ on $a < x_1 < \infty$.