  Based on recent progress in mathematical physics, we present a reliable
method to analytically solve the linearized BCS gap equation for a large class
of finite-range interaction potentials leading to s-wave superconductivity.
With this analysis, we demonstrate that the monotonic growth of the
superconducting critical temperature $T_c$ with the carrier density, $n$,
predicted by standard BCS theory, is an artifact of the simplifying assumption
that the interaction is quasi-local. In contrast, we show that any well-defined
non-local potential leads to a "superconducting dome", i.e. a non-monotonic
$T_c(n)$ exhibiting a maximum value at finite doping and going to zero for
large $n$. This proves that, contrary to conventional wisdom, the presence of a
superconducting dome is not necessarily an indication of competing orders, nor
of exotic superconductivity. Our results provide a prototype example and guide
towards improving ab-initio predictions of $T_c$ for real materials.
