Refined Limiting Profiles of the Principal Eigenvalue Problems with Large Advection

Abstract

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 $\hat{I}\text{c◯}\subset {\mathbb{R}}^N(N\ge 1)$ satisfying $\partial \hat{I}\text{c◯}\in C^2$ is a bounded domain and contains the origin as an interior point, the constants $\hat{I}\mu >0$ and $\hat{I}\pm >0$ are the diffusive and advection coefficients, respectively, and $m(x)\in C^2(\bar{\hat{I}}\text{c◯})$, $V(x)\in C^3(\bar{\hat{I}}\text{c◯})(0<{\hat{I}}^3<1)$ are given functions. We analyze the refined limiting profiles of the principal eigenpair $(\hat{I}»,\ddot{I}•)$ for (0.1) as $\hat{I}\pm \to \infty$, which display the visible effect of the large advection on $(\hat{I}»,\ddot{I}•)$. It expects that our argument is applicable to investigating the refined expansions of the general principal eigenvalue problems.