The convective eigenvalues of the one–dimensional p–Laplacian as p → 1

Abstract

This paper studies the limit behavior as p→1 of the eigenvalue problem{−(|ux|p−2ux)x−c|ux|p−2ux=λ|u|p−2u,0<x<1,u(0)=u(1)=0. We point out that explicit expressions for both the eigenvalues λn and associated eigenfunctions are not available (see [16]). In spite of this hindrance, we obtain the precise values of the limits limp→1+⁡λn. In addition, a complete description of the limit profiles of the eigenfunctions is accomplished. Moreover, the formal limit problem as p→1 is also addressed. The results extend known features for the special case c=0 ([6], [28]).