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} -\epsilon\Delta \phi-2\alpha\nabla m(x)\cdot\nabla \phi+V(x)\phi&=\lambda \phi\ \ \text{in}\ \ \Omega,\\ \phi&=0\ \ \hbox{on}\ \ \partial\Omega, ~~~\text{(0.1)} \end{split} \right. \end{equation} where $\Omega\subset\mathbb{R}^N~(N\geq1)$ satisfying $\partial\Omega\in C^{2}$ 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 $m(x)\in C^{2}(\bar{\Omega})$, $V (x)\in C^{\gamma}(\bar{\Omega})~(0<\gamma<1)$ are given functions. We analyze the refined limiting profiles of the principal eigenpair $(\lambda, \phi)$ for (0.1) as $\alpha\rightarrow\infty$, 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.