I am currently reading Del Pino & Dolbeault's paper http://capde.cmm.uchile.cl/files/2015/06/pino2002.pdf and I am trying to understand the proof of Theorem 4.
We assume that $1<p<\frac{d}{d-2}$ and we want to minimize the functional $$ G(w)=\frac{1}{2}\int_{\mathbb{R}^{d}}|\nabla w|^{2}dx+\frac{1}{p+1}\int_{\mathbb{R}^{d}}|w|^{p+1}dx, $$ subject to the integral constraint $$ \frac{1}{2p}\int_{\mathbb{R}^{d}}|w|^{2p}dx=J_{\infty}, $$ where $J_{\infty}$ is some constant (which is given but the exact value is not important for the proof).
The minimizer exists and satisfies the Euler-Lagrange equation $$ -\Delta w+w^p=\mu_{\infty}w^{2p-1}, $$ where $\mu_{\infty}$ is the Lagrange multiplier.
So far so good. Now the author claims that the Lagrange multiplier $\mu_{\infty}$ can be uniquely determined by the system $$ \frac{1}{2}\int_{\mathbb{R}^{d}}|\nabla w|^{2}dx+\frac{1}{p+1}\int_{\mathbb{R}^{d}}|w|^{p+1}dx=I_{\infty} $$ $$ \int_{\mathbb{R}^{d}}|\nabla w|^{2}dx+\int_{\mathbb{R}^{d}}|w|^{p+1}dx=2p\mu_{\infty}J_{\infty} $$ $$ \frac{d-2}{2d}\int_{\mathbb{R}^{d}}|\nabla w|^{2}dx+\frac{1}{p+1}\int_{\mathbb{R}^{d}}|w|^{p+1}dx=\mu_{\infty}J_{\infty} $$ where $I_{\infty}=\inf_{w\in\mathcal{X}}G(w)$ ($\mathcal{X}$ denotes the right space so that all the integrals above are well-defined). I understood how to obtain this system.
My first question is: Why is the Lagrange multiplier unique?
Moreover, the author uses the scaling $$ \overline{w}(x)=\lambda^{\frac{2}{p-1}}w(\lambda x) $$ to obtain that $\mu_{\infty}=1$. Could someone explain this step to me? I didn't understand this scaling argument.