I am studying the proof of the following theorem of this article
http://projecteuclid.org/download/pdf_1/euclid.cmp/1103922134
The theorem states:
Theorem: For $0 < \sigma < \frac{2}{N-2}$, let $$ \alpha = \displaystyle\inf_{u \in H^{1}(R^n)} J^{\sigma, N} (u)$$
Where $ J^{\sigma, N} (u) = \displaystyle\frac{|| \nabla u||^{\sigma N}_{L^2(R^n)} || u||^{2 + \sigma(2-N)}_{L^2(R^n)}}{|| u||^{2 \sigma +2}_{L^{2\sigma +2}(R^n)}}$.
The infimum above is attained at a function $\psi$ with the following properties
1) $\psi$ is a positive radial function
2) $\psi \in H^{1}(R^n) \cap C^{\infty}(R^n)$
3) $\psi$ is a solution of the equation
$$ \frac{\sigma N}{2} \nabla \psi - (1+ \frac{\sigma}{2} (2-N))\psi + \psi^{2 \sigma +1} =0$$
of minimal $L^{2}(R^n) $ norm . In addition,
$$ \alpha = \frac{||\psi||^{2 \sigma}_{L^2 (R^n)}}{\sigma +1}$$
I am not understanding the third part of the proof.
About the third part the autor writes :
Part (3) follows form the fact that $\psi^*$, the minimizing function is in $H^{1}(R^n)$ and satisfies the Euler Lagrange equation:
$$ \frac{d}{d \epsilon} J^{\sigma, N} (\psi^* + \epsilon \xi) = 0 , for \epsilon = 0, \forall \xi \in C^{\infty}_{0}(R^n) (*)$$
Taking into acount that $|| \psi^*||_{L^2(R^n)} = || \nabla \psi^*||_{L^{2}(R^n)} = 1$ we have
$$ \frac{\sigma N}{2} \nabla \psi^* - (1+ \frac{\sigma}{2} (2-N))\psi^* + \alpha (\sigma +1) {\psi^{*}}^{2 \sigma +1} =0 (**)$$
The smoothness of $\psi^*$ follow from a article of the referece. Let $\psi^* = [\alpha(\sigma +1)]^{-\frac{1}{2 \sigma}} \psi$. then $\psi$ satisfies the equation of the theorem and $\alpha = \frac{||\psi ||^{2 \sigma }_{L^{2}(R^n)}}{\sigma +1}$. this completes the proof of the theorem.
My questions:
1)If the function $h(\epsilon ) =J^{\sigma, N} (\psi^* + \epsilon \xi)$ is diferentiable with respect to the variable $\epsilon$ (with $\xi $fixed), then the diferential calculus tell us that the expression in $(*)$ is true. But how can i prove that $h(\epsilon)$ is diferentiable in any $\epsilon$?
2)How can i obtain $(**)$ form $(*)$ ? I dont see how to do that ..
3) how can i prove that $\psi$ is the solution of $L^2$ minimal norm ?
I dont know how to answer this questions ...
Someone can give me a help?
Any help will be apreciated.
thanks in advance
Let $I_1u=\|\nabla u\|_2^2$, $I_2u=\|u\|_2^2$ and $I_3u=\|u\|_{2\sigma+2}^{2\sigma+2}$. Note that $$\langle I'_1u,v\rangle=2\int\nabla u\nabla v,\ \ \langle I'_2u,v\rangle=2\int u v,\ \langle I'_3u,v\rangle=(2\sigma+2)\int |u|^{2\sigma+1}u v\tag{1}$$
If $u\neq 0$, we apply the quotient rule to conclude that $$\langle J^{\sigma,N}u,v\rangle=\frac{I_3u\langle[(I_1u)^{\sigma N/2}(I_2u)^{1+\sigma(2-N)/2}]',v\rangle-(I_1u)^{\sigma N/2}(I_2u)^{1+\sigma(2-N)/2}\langle I'_3,v\rangle}{(I_3u)^2}\tag{2}$$
Now you can apply the product and chain rule to (2) and then combine with (1) to get the desired result.