Actually, I'm reading a paper which finds the saddle point of a functional, of course the unbounded below energy functional will suggest a potential saddle, but the structure of mountain pass is the key point, if we have this structure, then we possibly have a saddle point. (The picture of the structure is showed as below.)
In the paper L.Martinazzi, there is a functional $$ I_\lambda(u)=\frac{1}{2} \int_M\left|\Delta_g^{\frac{m}{2}} u\right|^2 d \mu_g-\frac{\lambda}{2 m} \log \left(\int_M e^{2 m u} d \mu_g\right) $$ on a Banach space $E$， $$E:=\left\{u \in H^m(M): \int_M u d \mu_g=0\right\},$$ equipped with the norm $$ \|u\|:=\left(\int_M\left|\Delta_g^{\frac{m}{2}} u\right|^2 d \mu_g\right)^{\frac{1}{2}}. $$
its critical point is the famous mean-field equation $$\tag{1} \left(-\Delta_g\right)^m u+\lambda=\lambda \frac{e^{2 m u}}{\int_M e^{2 m u} d \mu_g} \text { on } M \text {. } $$
His theorem is that: Let $\lambda_1=\lambda_1(M)$ be the smallest eigenvalue of $\left(-\Delta_g\right)^m$ and $\Lambda_1:=$ $(2 m-1) ! \operatorname{vol}\left(S^{2 m}\right)$. Assume that $\Lambda_1 / \operatorname{vol}(M)<\lambda_1 /(2 m)$. Then for every $\lambda \in$ ]$\Lambda_1 / \operatorname{vol}(M), \lambda_1 /(2 m)\left[, \lambda \notin \frac{\Lambda_1 \mathbb{N}}{\operatorname{vol}(M)},(1)\right.$ has a non-constant solution.
The functional have a parameter $\lambda$, there is a constant $\Lambda_1$ such that when $\lambda>\Lambda_1$ then $I_\lambda$ is not bounded from below, which means that there always exists a function $u \in E$ such that $I_\lambda(u)<I_\lambda(0)=0$.
Besides, when $\lambda<\frac{\lambda_1(M)}{2 m}$, $I_\lambda(0)=0$ and $0$ is the strict local minimum (Could this create a crater?).
Now recall the mountain pass theorem:
Let $X$ be a Banach space, $I: X \rightarrow \mathbb{R}$ a $C^1$-functional satisfying the Palais–Smale condition and $I[0]=0$. Suppose
(A) $\exists \rho, \alpha>0$ such that $I[u]\geq \alpha$ if $\|u\|=\rho$,
(B) $\exists v \in X$ with $\|v\|>\rho$ such that $I[v] \leq 0$.
Then $I$ has a critical value $c \geq \alpha$ and $c$ is characterized by $$ c=\inf_{\gamma \in \Gamma} \max _{t\in [0,1]} I[\gamma(t)] $$ where $$ \Gamma=\{\gamma \in C([0,1], X) \mid \gamma(0)=0, \gamma(1)=v\} $$
Here, condition (A) is about the crater of our mountain pass, but for the paper I cited, $0$ being the strict local minimum seems to serve as the crater of the mountain pass, but in stackexchange, Lorenzo Pompili kindly gave me a counterexample to state that condition (A) is stronger than the condition that $0$ is the strict local minimum, I wonder if M.P.T is still applicable here. (The rest of their proof is the same as the requirement of M.P.T, they constructed $u_0$ s.t. $I\left(u_0\right)<0$ and $\left\|u_0\right\| \geq 1$, paths $P:=\left\{\gamma \in C^0([0,1] ; E): \gamma(0)=0, \gamma(1)=u_0, \gamma(t) \in C^{\infty}(M)\right.$ for $\left.0 \leq t \leq 1\right\}$, and $c_\mu:=\inf _{\gamma \in P} \max _{t \in[0,1]} I_\mu(\gamma(t))$, and finally proved the converging P-S sequence $u_n \rightarrow u$ with $u$ being the critical point of $I_\lambda$.)
Or did he just used the idea of M.P.T, not directly applied the M.P.T, since he independently proved the existence of the converging P-S sequence $u_n \rightarrow u$ and $u$ is a critical point of $I_\lambda$ without using the M.P.T.
