My questions arise from Here, it seems that I didn't give a clear question, so I rephrase my questions here.

For example, for $$ -\Delta u=f(u) \quad \text { in } \Omega, $$ we call a solution is stable if $$ Q_u(\varphi):=\int_{\Omega}|\nabla \varphi|^2 d x-\int_{\Omega} f^{\prime}(u) \varphi^2 d x \geq 0, \quad \forall \varphi \in C_c^1(\Omega) . $$ Here $Q_u(\varphi)$ is the second variation of its energy functional.

And the Morse index is defined by the maximal dimension of space $K_\varphi$ in which $Q_u(\varphi) <0$.

I have several questions.

1. Since the definition of stable solution is $Q_u(\varphi) \geq 0$, so based on the examples of the critical points for functions ($z=x^4-y^4$ on $p=(0,0)$, $p$ is a stable point but a saddle point; or, $y=-x^{100}$ on $\mathbb{R}$, $p=0$ is a stable point but a local maximum.), we can't say $u$ is a local minimum. If I didn't misunderstood, the place where ambiguity occurs is when $Q_u(\varphi) = 0$, for any $\varphi$. But I can't come up with any examples, is it possible to find a PDE whose solution satisfies $Q_u(\varphi) = 0$, for any $\varphi$?

2. If $Q_u(\varphi)<0$ strictly for any $\varphi$, will it be enough to show that $u$ is a local maximum? If so, if for any $\varphi \in C_c^1(\Omega), Q_u(\varphi)<0$, this means infinite Morse index, and $u$ is the local maximum?

3. I find an example of talking about the type of a critical point through calculating the second variation in this paper A note on Chern-Yamabe problem, let $(X, \omega)$ be a compact complex manifold of complex dimension $n$, they studied the following PDE $$-\Delta_\omega f+\mathrm{S}^{\mathrm{Ch}}(\omega)=\lambda \exp (2 f / n).$$ In Lemma3.3, they want to know what type of critical point is $f=0$ when $\mathrm{S}^{\mathrm{Ch}}(\omega) = \lambda$, so they give some specific direction $v$, then they have the second variation at $v$ as $$ \delta^2 \mathcal{F}(v)=\int_X\left(|d v|^2-\frac{2 \lambda}{n} \exp (2 f / n) v^2\right) d \mu . $$ Let $\lambda_1$ be the first principal eigenvalue of the Laplacian operator. Then their conclusion is If $\lambda_1 \geq 2 \lambda / n$, then $$ \delta^2 \mathcal{F}(v) \geq\left(\lambda_1-2 \lambda / n\right) \int_X v^2 d \mu \geq 0, \quad \forall v \text { with } \int_X v d \mu=0 $$ shows that $f=0$ is a local minimum. If $\lambda_1<2 \lambda / n$, then we can take some non-zero eigenvector $v_0$ with $\int_X v_0 d \mu=$ 0 and $$ \delta^2 \mathcal{F}\left(v_0\right) \leq\left(\lambda_1-2 \lambda / n\right) \int_X v_0^2 d \mu<0 . $$ Hence, $f=0$ is a saddle point and unstable.

Their conclusion made me more confused, the second variation $\ge 0$ directly implies local minimum? And they find a direction where the second variation $< 0$ then saddle point? Don't they need to find a direction where the second variation $> 0$? (Maybe through poincare inequality.)