The first nontrivial eigenvalue for a system ofp-Laplacians with Neumann and Dirichlet boundary conditions

Abstract

Abstract. We deal with the first eigenvalue for a system of two p−Laplacians with Dirichlet and Neumann boundary conditions. If ∆pw = div(|∇w| p−2 ∇w) stands for the p−Laplacian and α p + β q = 1, we considerwith mixed boundary conditions u = 0, |∇v| q−2 ∂v ∂ν = 0, on ∂Ω.

The first eigenvalue and eigenfunction of a nonlinear elliptic system

“…The existence of a principal eigenvalue, simplicity and the isolation of the first eigenvalue have been proved for (1.1) and its variants in [4,5,10,17,25]. Let us recall that the first eigenvalue λ(p, q) of (1.1) is simple if for any two pairs of corresponding eigenfunctions (u, v) and (φ, ψ) there exist real numbers k 1 and k 2 such that u = k 1 φ and v = k 2 ψ.…”

“…This system has been studied intensively by several authors, see e.g. [4,5,10,25] to list just a few references. The resonant quasilinear system (5.2) differs from (1.1) and has certain specific properties.…”

The problem of locating an obstacle for the fundamental eigenvalue of laplacian and p-laplacian

“…In [31] Leandro , Del Pezzo and Julio Studied the rst eigenvalue for the p-Laplacian operator with the boundary conditions of Dirichlet and Neumann (mixed boundary conditions). They considered the following problem:…”

Eigenvalues for systems of fractional $p$-Laplacians

“…On the other hand, there is a rich recent literature concerning eigenvalues for systems of p−Laplacian type, (we refer e.g. to [6,12,16,14,29] and references therein). The only references that we know concerning the asymptotic behaviour as p goes to infinity of the eigenvalues for a system are [5] and [12] where the authors study the behaviour of the first eigenvalue for a system with the usual local p−Laplacian operator.…”