Define an operator $L_{+}$ as follows: $$L_{+} = -\Delta + 1 - pQ^{p-1}$$ Let $Q$ be the solution to the nonlinear PDE: $$Q-\Delta Q - |Q|^{p-1}Q =0.$$ Let $$Q_1 = \left(\frac{2}{p-1} + x\cdot \Delta\right)Q.$$ I want to show that $L_{+}Q_1 = -2Q.$
Proof Attempt: We compute, $$-\Delta Q_1 = -\left(\frac{2}{p-1}\Delta Q + \Delta[x\cdot \Delta Q]\right)=-\frac{2}{p-1}\Delta Q-x\Delta^2Q-2\nabla Q-\Delta Q d.$$
$$-pQ^{p-1}(Q_1)=-pQ_1^{p-1} = -p\left(\frac{2Q}{p-1}+x\cdot \Delta Q\right)^{p-1}.$$
I am not sure whether I have understood how to apply the operators correctly to the functions given above and therefore I am stuck at this point. Perhaps someone give a few hints since this kind of computation is new to me.
Background: Start with the Nonlinear Schrodinger (NLS) equation: $$i\partial_t \psi =-\Delta \psi -|\psi|^{p-1}\psi$$ where $\psi(t,x):\mathbb{R}\times \mathbb{R}^{n}\to \mathbb{C}.$ Then we look for solutions called solitary waves that are of the form: $$\psi(t,x) = Q(x)e^{it}.$$ If you substitute this in the NLS you will get the PDE that I mentioned above.