I want to talk about generalizing the conception of 'stable' solution and 'stable outside the compact set' solution of elliptic PDE from $\mathbb{R}^n$ to closed manifold.

I'm reading $\Delta u +e^u=0$ in $\mathbb{R}^n$, it said that, a solution $u \in C^2(\Omega )$ of $-\Delta u=\mathrm{e}^u$ is stable in $\Omega$ if: $$ \forall \psi \in C_c^1(\Omega) \quad Q_u(\psi):=\int_{\Omega}|\nabla \psi|^2-\mathrm{e}^u \psi^2 \geqslant 0 . $$ Here $Q_u$ is the second variation of the energy function corresponding to the PDE.

What I'm thinking is that for a PDE on closed manifold, can I also calculate like this? (Compute the second variation of the energy functional), because the manifold itself is compact, so we don't need the test function with compact support and the classical solution of the PDE must be bounded. If we compute like this and get that there is no stable solution, can we say that the possible solution is the saddle point of the energy functional?

If we can do like this, here come another question, when discussing the finite Morse index solution, people always studying the 'stable outside the compact set' solution, the definition is: $$ \forall \psi \in C_c^1(\Omega \backslash \mathcal{K}) \quad Q_u \ge 0$$ here $\mathcal{K} \in \Omega$ and $Q_u $ is the second variation.

Because the finite Morse index solution is stable outside a compact set, but for closd manifold it's hard to define 'outside a compact set', so for Morse index not equal to 0, we can only study its Morse index by studying its negative eigenvalue, right?