Refined Limiting Profiles of the Principal Eigenvalue Problems with Large Advection

In this paper, we are concerned with the following eigenvalue problem with an advection term: \begin{equation}\label{0.1} \left{ \begin{split} -εΔφ-2α\nabla m(x)\cdot\nabla φ+V(x)φ&=λφ \ $\text{in}$\ \ Ω,\ φ&=0\ \ \hbox{on}\ \ \partialΩ, ~~~$\text{(0.1)}$ \end{split} \right. \end{equation} where $\mathrm{\Omega }\subset {\mathbb{R}}^N(N\ge 1)$ satisfying $\mathrm{\partial }\mathrm{\Omega }\in C^2$ is a bounded domain and contains the origin as an interior point, the constants $\epsilon >0$ and $\alpha >0$ are the diffusive and advection coefficients, respectively, and $m(x)\in C^2(\bar{\mathrm{\Omega }})$, $V(x)\in C^{\gamma }(\bar{\mathrm{\Omega }})(0<\gamma <1)$ are given functions. We analyze the refined limiting profiles of the principal eigenpair $(\lambda ,\phi )$ for (0.1) as $\alpha \to \mathrm{\infty }$, which display the visible effect of the large advection on $(\lambda ,\phi )$. It expects that our argument is applicable to investigating the refined expansions of the general principal eigenvalue problems.