I initially posted this question on Stackexchange Mathematics (see here) but as I got no answer, maybe someone here will be able to help me.
I am watching a serie of lectures on "Blow up solution for the energy critical heat equation" from Monica Musso on YouTube and at some point she states a result I do not understand. Let met recall the setting.
We are studying the following system: $$u_t = \Delta u + u^{N+2/N-2} \quad \text{in }\Omega \times (0,T), \quad \text{and} \quad u = 0 \quad \text{on } \partial \Omega \times (0,T).$$ We also require $u(\cdot, 0) = u_0$ in $\Omega$. Steady states are solutions of the problem that do not depend on time, i.e. they satisfy $$0 = \Delta u + u^{N+2/N-2}.$$ We may show that, steady states are given by $$U_{\lambda, \xi}(x) = \frac{1}{\lambda^{N - 2/2}}U\left(\frac{x - \xi}{\lambda}\right), \quad\text{where } U(y) = \frac{\alpha_N}{(1 + |y|^2)^{N - 2/2}}.$$ We linearized this equation around $U$ and we define $$L(\varphi) = \Delta \varphi + \frac{N + 2}{N - 2} U^{\frac{N+2}{N-2} - 1}\varphi.$$ At this point, she claimed that, by Noether's theorem, all the invariances of the solutions of the initial problem will generate an element of the kernel of the operator $L$, namely $$\text{span}\left(\partial_1 U, \ldots, \partial_N U, \frac{N - 2}{2} U + \nabla U \cdot y\right) \subset \text{ker }L,$$ where the first $N$ elements come from the invariance by translation and the last one comes from the scaling invariance. Moreover, we may show that this kernel is in fact non-degenerate, in the sense that we have the equality instead of the inclusion.
My question is about this use of Noether's theorem. I tried to read its statement in Lee's Introduction to Smooth Manifold, but it was really differential geometry oriented and I admit that I do not totally get the link with the above problem. Could one of you try to explain it to me ? Or at least giving me a reference with a more "Analysis-version" of this theorem ? I have some background in differential geometry but I am not totally comfortable with it.