In a given domain $\Omega$, we have: $\Delta u=-\lambda u$ with $u>0$. My question is: does this mean that $u$ is principal eigenfunction for $\Delta$ in $\Omega$?

Also, more generally, does this also apply for $Lu=a^{ij}u_{ij}+b^iu_i+cu$?

I tried to prove it by using variational way through considering $\frac{\int|Du|^2}{\int u^2}$, but didn’t figure out if it’s right.

Also, does the area of $\Omega$ have any relationship with the principal eigenvalue of $L$? I know we can give a lower bound by the principal eigenvalue of $\Delta$ and wish to find something similar for $L$.