On the local boundedness of solutions to the equation $-\Delta u+a\partial_zu=0$

Equation $-\Delta u+a\partial_zu=0$ is considered in a domain in $n$-dimensional space. The coefficient in a minor term does not depend on the direction of differentiation in this term. For $a\in L_p$ with $p>\frac{n-1}2$ it is proven that a solution $u$ is locally bounded. If $p=\frac{n-1}2$ then a solution can be unbounded.