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 $\alpha\in{\mathbb N}_0^N$, $$ (\partial^\alpha u)(x)= O(e^{-\varepsilon\,|x|}) \enspace \text{as $|x|\to\infty$.} $$

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 http://link.springer.com/article/10.1007%2Fs11854-010-0021-4.

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 $\alpha\in{\mathbb N}_0^N$, $$ (\partial^\alpha u)(x)= O(e^{-\varepsilon\,|x|}) \enspace \text{as $|x|\to\infty$.} $$

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 $\alpha\in{\mathbb N}_0^N$, $$ (\partial^\alpha u)(x)= O(e^{-\varepsilon\,|x|}) \enspace \text{as $|x|\to\infty$.} $$

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 $\alpha\in{\mathbb N}_0^N$, $$ (\partial^\alpha u)(x)= O(e^{-\varepsilon\,|x|}) \enspace \text{as $|x|\to\infty$.} $$