$$\beex \bea \int \lap f|f|^{q-2}f\rd x &=-\int \n f\cdot \sez{(q-2)|f|^{q-3}\cfrac{f}{|f|}\n f\cdot f +|f|^{q-2}\n f}\rd x\\ &=-\int (q-2)|f|^{q-4}|f|^2|\n f|^2 +|f|^{q-2}|\n f|\rd x\\ &=-(q-1)\int |f|^{q-2}|\n f|^2\rd x\\ &=-(q-1)\int |f|^{q-2}|\n |f||^2\rd x\quad\sex{\n|f|=\cfrac{f}{|f|}\n f}\\ &=-(q-1)\int | |f|^{\frac{q}{2}-1}\n |f| |^2\rd x\\ &=-\cfrac{4(q-1)}{q^2} \int| \n |f|^{\frac{q}{2}} |^2\rd x. \eea \eeex$$