Consider the Hénon problem in the ball:
\begin{equation} \left\{ \begin{array}{rcll} - \Delta u & = & |x|^\alpha u^{p - 1} & \quad \text{ in } B \\ u & > & 0 & \quad \text{ in } B \\ u & = & 0 & \quad \text{ on } \partial B \end{array} \right. \end{equation} where $2 < p < 2^*$.
My book (Morse Index of Solutions of Nonlinear Elliptic Equations, by Damascelli and Pacella) claims that there exists a solution which is the minimizer of $$ S_\alpha = \inf_{v \in H_0^1(B), v \not\equiv 0} \frac{\int_B |\nabla v|^2 \ \mathrm{d}x}{(\int_B |x|^\alpha v^p \ dx)^{2/p}}. \label{1}\tag{*} $$
Now, as I understand, we indeed have that there is a solution $u$ that is the minimizer of $$ J(u) = \inf_{\frac 1 p \int_B |x|^\alpha v^p \, \mathrm{d}x = 1/p} \int_B |\nabla v|^2 \ \mathrm{d}x \label{2}\tag{**} $$
My question is,
Why are \eqref{1} and \eqref{2} equivalent?
Thanks in advance.
David Stolnicki helped me:
Upon setting $a(v) = \frac{1}{(\int |x|^\alpha v^p)^{1/p}}$, we have $$ \inf_{v \neq 0} \frac{\int |\nabla v|^2}{(\int |x|^\alpha v^p)^{2/p}} = \frac{a^2}{a^2} \inf_{v \neq 0} \frac{\int |\nabla v|^2}{(\int |x|^\alpha v^p)^{2/p}} = \inf_{\int|x|^\alpha w^p = 1} \int |\nabla w|^2 $$
As $v$ runs through $H_0^1$, $w$ runs through the restriction and we are done.