Let $\Omega\subset\mathbb{R}^d$ a connected open set (which is not necessarily bounded). Assume that $f\in C_0^\infty(\Omega)$ with $\operatorname{supp}(f)\subset K,$ with $K$ compactand let $u$ be a solution to the equation $$-\Delta u+u=f\quad\text{in }\Omega,\\ \quad\quad u\in H^1_0(\Omega).$$ I'm looking for a statement of the following type: For every $\epsilon>0$ there exists a $R>0$ (depending only on $K$ but not on $f$) such that $$\|u\|_{L^2(\Omega\setminus B_R(0))}\leq \epsilon\|f\|_{L^2}.$$
This is easily obtained for $d=3$, $\Omega=\mathbb{R}^3$ using the fundamental solution, but the general case is not clear to me...