For the computation of the effective action in section 11.4 Peskin & Schroeder choose the perturbative expansion of the generating functional. For this purpose they split the full Lagrangian up in two parts:
$$ {\cal L} = {\cal L}_1 + \delta {\cal L} \tag{11.54}$$
where the first part is the Lagrangian part depending on the renormalized parameters and the second part consists of the counterterms. Then in the following an at least implicitly given relationship between the classical field $\phi_{cl}$ and the external source $J(x)$ has to be found.
At lowest order of perturbation theory that relation is just the classical field equation:
$$ \frac{\delta {\cal L}}{\delta \phi}|_{\phi = \phi_{cl}} + J(x) = 0. $$
If I interpret this equation correctly it should yield for the $\phi^4$ theory in something like
$$[(\Box +m^2)\phi + \lambda\frac{\phi^3}{3!}]\mid_{\phi=\phi_{cl}} + J(x) = 0$$
Or is the field equation according to the Lagrangian(10.16) meant ?
$$[Z(\Box + m_0^2)\phi_r + \lambda_0 Z^2\frac{\phi_r^3}{3!}]\mid_{\phi_r = \phi_{cl}} + J(x) =0$$
So if in a second step I develop this equation as
$$[(\Box + m_0^2)\phi_r + \lambda_0 \frac{\phi_r^3}{3!} + (\delta_Z \Box + \delta_m)\phi_r + \delta_{\lambda} \frac{\phi_r^3}{3!}]\mid_{\phi_r = \phi_{cl}} + J(x) = 0 $$
However, what does it mean a field equation with counter terms in lowest order --- i.e. zeroth order ? No counter terms?
Then they define another function $J_1(x) $ which is supposed to fulfill this equation exactly when ${\cal L} = {\cal L}_1$:
$$ \frac{\delta {\cal L}_1}{\delta \phi}|_{\phi = \phi_{cl}} + J_1(x) = 0. \tag{11.55}$$
So according to the formula (10.18) mentioned in the paragraph before equation (11.54) of P&S I would guess that it would mean:
$$[(\Box +m^2)\phi_r + \lambda\frac{\phi_r^3}{3!}]\mid_{\phi_r=\phi_{cl}} + J_1(x) = 0$$
Actually, the guessed equations all look very similar, actually I don't really understand what is meant for a field equation to be in lowest order OR exactly.
What means this distinction (lowest order OR exactly) and what is the underlying reason for this distinction? I guess, I miss something more profound here.
I also wonder about the functional variation of the Lagrangian, I guess, P&S rather means the functional variation of the action: $\frac{\delta S}{\delta \phi}$. Is this right? Or am I completely mislead?
