On eigenvalue problems of 𝑝-Laplacian with Neumann boundary conditions

Abstract

Abstract.We study the nonlinear eigenvalue problem -Au = Xm(x)\uf~~u iniî, -=0 onc*C2, where p > 1 , À e R. p OnFor fn m(x) < 0, we prove that the first positive eigenvalue À, exists and is simple and unique, in the sense that it is the only eigenvalue with a positive eigenfunction.In the case jnm(x) = 0 , we prove that A0 = 0 is the only eigenvalue with a positive eigenfunction.

Variational and non-variational eigenvalues of the p-Laplacian

“…We will show that the variational eigenvalue μ 0 constructed above is the minimal (Carathéodory) eigenvalue of (2.1)-(2.3), and that it is simple and is the unique principal eigenvalue (that is, whose eigenfunctions do not change sign). Similar results have been proved for Dirichlet and Neumann problems in, for example, [1,11,15,16]. On the other hand, the minimal periodic eigenvalue does not seem to have been treated before for general L 1 coefficients.…”

Existence of solutions of the Neumann problem for a class of equations involving the p -Laplacian via a variational principle of Ricceri

“…Many papers deal with the eigenvalue problems related to the p-Laplacian with Neumann conditions. We refer for instance to [4] and to the references therein for details. The existence of multiple or positive solutions is also widely investigated [2,7,5], but for 1 < p < N and, usually, for specific nonlinearities.…”

On the spectrum of one dimensional p-Laplacian for an eigenvalue problem with Neumann boundary conditions

“…[1,6,15,13,7,8,3,10,5,11,2,12,16,4,9] and the references therein). In particular, we present here some results that motivated this work: First, we mention the result in [8], that we consider as a principal key for the development of our results.…”