This might be very trivial, or not.
Let $p\colon\mathbb{R}^n\to \mathbb{R}$ be a polynomial of even degree, at most $n-2$. Assume that $p(x)\leq 0$ for any $x\in\mathbb{R}^n$. Assume that there exists $s\geq 0$ such that
\begin{equation} \int_{\mathbb{R}^n}|x|^s e^{p(x)}\, dx<\infty. \end{equation}
The question is if it is then possible to find $\varepsilon>0$ such that \begin{equation} \int_{\mathbb{R}^n}|x|^{s+\varepsilon} e^{p(x)}\, dx<\infty. \end{equation}
In other words: if the set of $s\geq 0$ such that $$ \int_{\mathbb{R}^n}|x|^s e^{p(x)}\, dx<\infty $$ is non empty, is it also open?