About function satisfying an identity involving its $L_2$ norm

Question

I want to proof that if $N\geq 3$ the equation $-\nabla^2f(x)=3f(x)$ only admits the trivial solution in $L^2(\mathbb{R}^N)$. I know that the solutions of this equation satisfy the identity $(N-2)\||\nabla f|\|_2^2=3N\|f\|_2^2$. Could someone give a counterexample or a proof for this fact?

Answer 1

Fix $f$ such that the quantities in the equation are finite and non-zero, and test the equation on $f_{\lambda}(x):=f(\lambda x)$ for $\lambda \in \mathbb{R}$.

Since the left-hand side scales as $|\lambda|^{2-N}$ and the right-hand side as $|\lambda|^{-N}$, you can always find a value of $\lambda\neq 0$ for which the equation holds.

As a matter of fact, the only way you can make a statement of this form work is by replacing e.g. the $L^2$ norm on the right-hand side with the $L^{2^*}$ norm where $2^*$ is the critical Sobolev exponent. Then you will get the correct scaling.