I was wondering if there is a "Laplace's Method" to estimate, as $n \to \infty$, integrals of type $$ I_n = \int_0^\infty e^{nh(x)}g(nx) \, dx $$ where $g$ is a smooth function, that converges to a limit $L$ when $x \to \infty$. Function $h$ achieves its maximum at some point $x_0 > 0$ and it is twice differentiable.
Alternatively, it would be useful for me a method for the integral $$ I_n = \int_0^\infty e^{nh(x, n)} g(x) \, dx. $$
Thanks in advance