In the standard reference of Gilberg and Trudinger there is an example of the function $u(x,y)=(x^2-y^2)\log^{1/2}\left(\frac1{x^2+y^2}\right)$ s.t. $\Delta u$ is continuous in some neighborhood of the origin, but $u$ does not belong to $C^2$ there. Adding some solution of а homogeneous equation would't help. So $f=\Delta u$ is not in the image of the mapping in question.

In the standard reference of Gilbarg and Trudinger there is an example of the function $u(x,y)=(x^2-y^2)\log^{1/2}\left(\frac1{x^2+y^2}\right)$ s.t. $\Delta u$ is continuous in some neighborhood of the origin, but $u$ does not belong to $C^2$ there. Adding some solution of а homogeneous equation would't help. So $f=\Delta u$ is not in the image of the mapping in question.