My teacher tell me that the eigenvalue question $$ \left\{ \begin{aligned} & -\Delta u=\lambda u ~~~x\in \Omega \\ & ~~~ u=0 ~~~~~~~~~~~x\in\partial\Omega \end{aligned} \right. $$ has been resolved . All eigenvalue such that $0<\lambda_1<\lambda_2... $ and $\lambda_k \sim C_n(\frac{k}{|\Omega|})^{2/n}$, and $\lambda_1(\Omega)\ge\lambda_1(B) \text{ when }~|\Omega|=|B|$ and $B$ is ball. And ...... (Sorry, I know it's very inaccurate).
I want to know which paper contains above content.
This article contains the derivation of some basic facts. The growth of the $\lambda$ to $\infty$ is shown, and if you look closely at the proof you also get an estimate, but not as good as you have asked for I think.
On this page you'll find references to more special results, including the growth estimate you've asked about.