For an elliptic operator
$$ Lu = (a^{ij} D\_iD\_j + b^i D\_i + c)u = f,$$
with suitable assumptions on the coefficients, one usually has Schauder estimates of the form
$$ \|u\|\_{C^{2, \alpha}} \leq C(\|f\|\_{C^{0, \alpha}} + \|u\|\_{L^\infty}).$$
I am interested in the case $\alpha =0$. It seems that the corresponding statement is generally wrong in this case.


For example, is the operator $-\Delta + V$ not surjective on $C^0(\Omega)$? (Where I assume $V \geq \varepsilon > 0$).


I am looking for references to this topic; most sources only discuss the case $\alpha>0$.



Estimate for elliptic problem on continuous functions
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. 


