Let $\Omega\subset\mathbb{R}^N$ be an open bounded domain. Let $u\in H^1_0(\Omega)$ and consider $$g(u):= e^{\alpha u^2}-1,$$ with $\alpha>0$. As an exercise, it is left to evaluate $$(g(su), u)\quad u\in H^1_0(\Omega)$$ where $(\cdot, \cdot)$ denotes the inner product in $H^1_0(\Omega)$ and $s\ge 0$.
This is my solution: Since $(u, v)=\int_{\Omega} \nabla u\cdot\nabla v dx$ for any $u, v\in H^1_0(\Omega)$, it is $$(g(su), u)=\int_{\Omega}\nabla (e^{\alpha s^2 u^2}-1)\cdot \nabla u dx =\int_{\Omega}2\alpha s^2 u e^{\alpha s^2 u^2} \nabla u\cdot\nabla u dx =2\alpha s^2 \int_{\Omega}u e^{\alpha s^2 u^2} |\nabla u|^2 dx.$$
Could someone please tell me if it is true?
Thank you in advance!