Let $u$ be a positive function on $\mathbb R^n$ such that
$$
\Delta u-\partial\_{x\_1}u=0,
$$
where $\Delta$ is the Laplacian operator $\partial\_{x\_1}^2+\partial\_{x\_2}^2+\cdots+\partial\_{x\_n}^2$.


Can we prove that $u=c\_1e^{x\_1}+c\_2$ for some constants $c\_1 \ge 0$ and $c\_2 \ge 0$?



The positive solutions of the weighted Laplacian equation
The function $u(x) = e^{a \cdot x}$ is a positive solution for any vector $a$ in the sphere $\partial B\_{1/2}\left(\frac{e\_1}{2}\right)$. So is $u = e^{-\frac{x\_1}{2}} w$ for any positive solution $w$ to $\Delta w - \frac{1}{4}w = 0$ (e.g. a radial one and any of its translations), and any (positive) linear combination of the above. 


