Why do I need to multiply by the function w in the energy method to guaranty at most one solution?
This is the example $\lambda u- \Delta u=f(x) \hspace{1cm} x \in \Omega, \hspace{1cm} u=0 \hspace{0.1cm} \in \partial \Omega)$
I am applying the energy method: For this, I suppose there are two solutions $u$ and $v$ and construct $w=u-v$. In the process, I multiply $w$ by the whole expression i.e.
$\lambda w^2- w\Delta w=0$
After some calculation, I check that $u=v$. But, where is the explanation of the method that explains that it is necessary to multiply by w?