Sharp estimates of the generalized principal eigenvalue for superlinear viscous Hamilton–Jacobi equations with inward drift

Abstract

This paper is concerned with the ergodic problem for viscous Hamilton–Jacobi equations with superlinear Hamiltonian, inward-pointing drift, and positive potential function which vanishes at infinity. Assuming some radial symmetry of the drift vector field and the potential function outside a large ball, we obtain sharp estimates of the generalized principal eigenvalue with respect to a perturbation of the potential function. We also specify the necessary and sufficient condition so that the spectral function contains a plateau.