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 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.
