I am interested in solutions $u(x)$ of the following equation: $$ -\Delta u(x) = f(x) \qquad R^N, $$ where we assume that $ u(x) \rightarrow \infty$ as $ |x| \rightarrow \infty$.
I am interested in obtaining estimates of the form
$$ \sup (1+|x|^t ) |u(x)| \le C ||f||'$$
where $ t>N-2$ and where $ \| \cdot \|$ is some appropriate norm.
Note that this will probably require some cancellation in $f$ since if $ f $ is nonnegative and compactly supported then using the Newtonian potential expression for $u(x)$ one sees that $ u(x)$ is roughly $ |x|^{2-N}$ for large $x$.
Thanks in advance.
