I am a student studying Michael Struwe's Variational method book. In this book, he said using the homogeneity of $$-\Delta u+\lambda u=u|u|^{p-2},$$ a solution of problem $$\begin{array}{lcl} -\Delta u+\lambda u=u|u|^{p-2} \mbox{ in } \Omega\\ u>0 \mbox{ in } \Omega\\ u=0 \mbox{ on } \partial \Omega \end{array} $$ can be obtained by solving a constrained minimization problem for the functional $$E(u)=\frac{1}{2}\int_\Omega{|\nabla u|^2+\lambda|u|^2 dx}$$ on $H_0^{1,2}(\Omega)$, restricted to the set $$M=\{u\in H^{1,2}_0(\Omega);\int_\Omega {|u|^p dx}=1\}.$$
What I can't understand is, firstly, I couldn't get why this equation is homogeneous equation. I think if I multiply a constant to u, the left side of equation is homogeneity 1 and the right side is homogeneity p-1, so it doesn't fit.
The second question is, if I accept the equation is homogeneous, how that fact can be used in this process.
I got everything else like how to make such E(u) form, but because of these questions, I feel that I can't fully understand it.
Thank you for reading my question. If you know the reason, please help me!