The small diffusion limit of the principal eigenvalue problems with advection

This paper is concerned with the following second order principal eigenvalue problem with an advection term:$-\epsilon \Delta \varphi -2\alpha \nabla m\left(x\right)\cdot \nabla \varphi +V\left(x\right)\varphi ={\lambda }_{\epsilon }\varphi \text{in}{H}_0^1\left(\Omega \right),$ where $\Omega \subseteq R^N(N\ge 1)$ is a bounded domain with smooth boundary ∂Ω and contains the origin as an interior point, the constants $\epsilon >0$ and $\alpha >0$ are the diffusive and advection coefficients, respectively, and $m\left(x\right)\in C^2\left(\overline{\Omega }\right)$, $V\left(x\right)\in C^{\gamma }\left(\overline{\Omega }\right)(0<\gamma <1)$ are given functions. We investigate the refined limiting profiles of the principal eigenpair for the above eigenvalue problem in the small diffusion limit (i.e., $\epsilon \to 0^{+}$), where the advection term $m\left(x\right)$ can be degenerate.