Given a PDE, $$-\Delta Q+Q-|Q|^{p-1}Q=0, \quad x \in \mathbb{R}^{N}$$ for $p>2$ we consider positive radial solutions of the form $Q(x) = q(|x|).$ Then it can be shown that, $$q^{\prime \prime}+\frac{N-1}{r} q^{\prime}-q+q^{p}=0, \quad q^{\prime}(0)=0, \quad \lim _{r \rightarrow \infty} q(r)=0$$ and for some constant $\kappa >0$ and all $r>1$, $$\left|q(r)-\kappa r^{-\frac{N-1}{2}} e^{-r}\right|+\left|q^{\prime}(r)+\kappa r^{-\frac{N-1}{2}} e^{-r}\right| \lesssim r^{-\frac{N+1}{2}} e^{-r}.$$ Now I want to show two things,
- $\int \partial_{x_i} Q(x) \partial_{x_j} Q(x) dx = 0$ for all $i\neq j.$
- $\int \partial_{x_i}\partial_{x_k} Q(x) \partial_{x_j}Q(x) dx = 0$ for all $i,j,k.$
For 1. I tried to re-write $$\int \partial_{x_i} Q(x) \partial_{x_j} Q(x) dx = \omega_{N-1}\int \frac{x_ix_j}{r^2} [q'(r)]^2 r^{N-1} dr \\ =\omega_{N-1}\int \frac{x_ix_j}{r^2} [q'(r)]^2 r^{N-1} dr$$ where $\omega_{N-1}$ denotes the surface area of $(N-1)$ dimensional sphere but I am not sure how to proceedafter this step.
For 2. I tried integration by parts, but I end with the same expression with different indices.
Any suggestions/comments about how to prove these two statements will be much appreciated.