Consider the following integral $$ I(s)=\int_{0}^{\infty}{J_{\frac{n-2}{2}}}(sr)r^{A+1}(e^{-r^{2\alpha}}-1)dr, $$ where $J_{\frac{n-2}{2}}$ is the Bessel function of order ${\frac{n-2}{2}}$, $s, A, \alpha>0$. I want to know the asymptotic property of $I(s)$ when $s\to+\infty$. Especially, I expect that we have $I(s)=o(s^{-\frac{n+2}{2}-A})$.
I think the way to estimate such kinds of integrals is stationary phase. Can someone show me how to deal with it? Thanks very much