I have a question regarding an equation in the book "Gauge/Gravity Duality" by Martin Ammon and Johanna Erdmenger which however can also be found in other AdS/CFT books or lecture notes.
The authors motivate that the "classical" AdS/CFT correspondence (${\cal N}=4$ SYM <-> IIB superstring theory on $AdS_5 \times S^5$) equates the partition function of the string theory with the generating functional of the CFT correlation functions. In the weak limit, i.e. for $N \rightarrow \infty$ and large $\lambda$, $N$ being the rank of the $SU(N)$ gauge group and $\lambda$ being the 't Hooft coupling, the explicit relation is given in terms of a saddle point approximation
$$\langle \text{exp} \big( \int d^d x \, \mathcal{O} \phi_{(0)} \big) \rangle_{CFT} = e^{-S_{sugra}} \vert_{\lim \limits_{z \to 0}(\tilde{\phi}(z,x) z^{\Delta-d})= \phi_{(0)}(x)} \tag{1}.$$
It is stated that $\tilde{\phi}$ denotes the solution of type IIB supergravity with leading asymptotic behaviour $z^{d−\Delta}\phi_{(0)}$ near the conformal boundary at $z = 0$.
I am a novice in supergravity and AdS/CFT which is why I might lack the required background to fully understand equation (1) and which gets revealed in the following questions:
Is $\tilde{\phi}$ an abbreviation for the collection of field solutions of the respective equations of motions obtained from the variation of the action w.r.t each field that occurs in SUGRA, $\frac{\partial S_{sugra}}{\partial \phi_{i}} = 0$? I.e. $\tilde{\phi} = \{ \phi_1,...,\}$ (by the way, what is the field content of SUGRA?) And what use does it have to plug back the solution of the equation of motions into the action that has been used to retrieve it?
And if this is the case: Can we always find solutions that satisfy the boundary behaviour, i.e. that have the property $$\lim \limits_{z \to 0} \tilde{\phi}(z,x)= z^{d-\Delta} \phi_{(0)}(x)~?$$