From the Matlab numerical experiments, I could observe:
$$I(\theta)\equiv \frac{I_1(\theta)}{I_2(\theta)}\equiv \frac{\int_0^{\infty}x^k e^{\theta x-x^{1+k}}dx}{\int_0^{\infty}e^{\theta x-x^{1+k}}dx}$$
converges to a finite limit as $\theta\rightarrow \infty $ whenever $0<k<1$, whereas $I(\theta)$ diverges to infinity as $\theta\rightarrow \infty $ when $k\geq 1$. I really want to verify this observation analytically.
Notice that $I(\theta)=E[X^k]$, where $X$ has a probability density $$\frac{e^{\theta x-x^{1+k}}}{\int_0^{\infty}e^{\theta x-x^{1+k}}dx}$$ on the support $[0,\infty)$.
When $k=1$, $I(\theta)=E[X]$ is an expected value of truncated normal random variable on $[0,\infty)$. In particular, when $0<k<1$, the density decays slower than the normal density function.
Any help would be very much appreciated! Thank you.