I am interested in the critical equation $- \Delta w(x) = w(x)^p $ in $ R^N$ where $p=\frac{N+2}{N-2}$. After translation the solutions of this equation are all radial with maximum at the origin (and there is an explicit formula). Lets assume $ w(x)>0$ is the radial solution of the above problem with the added condition that $w(0)=1$.

Define $ L(\phi)= - \Delta \phi - p w(x)^{p-1} \phi$. My interest is in the kernel of $L$ and lets not worry about the exact domain of $L$.

So we expect that $ \phi_i:=w_{x_i}$ is in the kernel of $L$ for each $ 1 \le i \le N$. But also note the original equation is scale invariant and lets assume that $ w_\lambda(x)$ satisfies the equation (without the constraint $w_\lambda(0)=1$)) for all $ \lambda >0$ with $ w_1(0)=1$. Then $ \phi_{N+1}:=\partial_\lambda w_\lambda(x)|_{\lambda=1}$ is also in the kernel of $L$.

So my question is: is the kernel of $L$ given by $ \{\phi_i : 1 \le i \le N\}$ or $ \{ \phi: 1 \le i \le N+1\}$ or neither.

This answer is well known (just not by me and I haven't been able to locate the answer). I attempted to see if $\phi_{N+1}$ is really a linear combination of the other terms (recall there is an explicit formula for $w(x)$ but I am getting bog down in computations).

thanks