Limiting behavior of principal eigenvalues and eigenfunctions for a class of elliptic operators with degenerate large advection

In this paper we study, both numerically and analytically, the asymptotic behavior of the principal eigenfunction of \eqref{1.1}, normalized by \eqref{1.2}, as $s\uparrow +\infty$. Based on the numerical computations of this paper, we can prove that, under condition (Hm) bellow, $\varphi_s$ approximates $1$ and $\varphi_s'$ approximates $0$, uniformly in $[-1,1]$, as $s\uparrow +\infty$. As a byproduct of this result, we can derive the asymptotic behavior of the principal eigenvalue in a one-dimensional situation not previously covered by \cite{ChLo} and \cite{PeZh}, as we are working under minimal regularity assumptions on $m(x)$. A recent result of \cite{BWZ} shows that the principal eigenvalue might oscillate as $s\uparrow +\infty$ if $m(x)$ is highly oscillatory. Thus, the oscillatory and regularity properties of $m(x)$ might severely affect the asymptotic behavior of $(\lambda_s,\varphi_s)$ as $s\uparrow +\infty$.