I am now reading the paper (or review for beginners), A bird's-eye view of density-functional theory, but I could not understand that the energy minimization problem, in which the derivative of the kinetic energy functional can not be written, can be replaced by solving the Kohn-Sham equation (page 33).
First, the energy minimization problem is written as follows:
\begin{eqnarray} 0 = \frac{\delta E}{\delta \rho(\mathbf r)} = v_\text{ext}(\mathbf r) + v_\text{H}(\mathbf r) + \frac{\delta T_\text{s}[\{\psi_n\}]}{\delta \rho(\mathbf r)} + v_\text{xc}(\mathbf r), \end{eqnarray}
where the external and Hartree terms, $v_\text{ext}$ and $v_\text{H}(\mathbf r)$ are written exactly, and $v_\text{xc}$ can be calculated explicitly once an approximation, such as the local density approximation (LDA) and generalized gradient approximation (GGA), is determined.
Here, because the single-particle exact kinetic energy, $T_\text{s}[\{\psi_n\}]$, is written as an orbital functional (not a density functional $T_\text{s}[\rho]$), we can not write the derivative form.
Subsequently, the paper describes that Kohn and Sham replace this problem by solving an equation (of course, this is the Kohn-Sham equation).
However, there is a gap in my understanding here. I understand the form of the energy minimization problem, and I also understand that the functional derivative of $T_\text{s}[\{\psi_n\}]$ with respect to $\rho$ cannot be written exactly. However, why are these two problems equivalent to solving the KS equation, that is, the eigenvalue problem?