There has recently been work by M. Cappiello, T. Gramchev, and L. Rodino on related problems, see e.g.  *Entire extensions and exponential decay for semilinear elliptic equations*, J. Anal. Math. **111** (2010), 339-367. 

An exemplary result reads: Suppose that the nonlinearity $f$ is a polynomial in $u$ and that $u\in \langle x \rangle^\delta H^s({\mathbb R}^N)$, where $s>N/2$,
$\delta>0$, is a solution of the semilinear elliptic equation shown above. Then $u$ belongs to the Gelfand-Shilov space $S_1^1({\mathbb R}^N)$. Amongst others, this implies that there is an $\varepsilon>0$ so that,  for all $\alpha\in{\mathbb N}_0^N$, 
$$
  (\partial^\alpha u)(x)= O(e^{-\varepsilon\,|x|}) \enspace \text{as $|x|\to\infty$.}
$$