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 { − ( | u x | p − 2 u x ) x − c | u x | p − 2 u x = λ | u | p − 2 u , 0 x 1 , u ( 0 ) = u ( 1 ) = 0 . We point out that explicit expressions for both the eigenvalues λ n and associated eigenfunctions are not available (see [16] ). In spite of this hindrance, we obtain the precise values of the limits lim p → 1 + ⁡ λ n . 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] ).