The paper considers the Caputo elliptic equation with nonlocal condition. We obtain the upper bound of the mild solution. The second contribution is to provide the lower bound of the solution at terminal time. We show the convergence results between the Caputo-modified Helmholtz equation and the Caputo Poisson equation. The main tool is the use of upper and lower bounds of the Mittag-Leffler function, combined with analysis in Hilbert scales space.