Steklov problem with an indefinite weight for the p-Laplacian
Date
2005-08-14
Authors
Torne, Olaf
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University-San Marcos, Department of Mathematics
Abstract
Let Ω ⊂ ℝN, with N ≥ 2, be a Lipschitz domain and let 1 < p < ∞. We consider the eigenvalue problem ∆2u = 0 in Ω and |∇u|p-2 ∂u/∂v = λm|u|p-2u on ∂Ω, where λ is the eigenvalue and u ∈ W1,p(Ω) is an associated eigenfunction. The weight m is assumed to lie in an appropriate Lebesgue space and may change sign. We sketch how a sequence of eigenvalues may be obtained using infinite dimensional Ljusternik-Schnirelman theory and we investigate some of the nodal properties of eigenfunctions associated to the first and second eigenvalues. Amongst other results we find that if m+ ≢ 0 and ∫∂Ωmdσ < 0 then the first positive eigenvalue is the only eigenvalue associated to an eigenfunction of definite sign and any eigenfunction associated to the second positive eigenvalue has exactly two nodal domains.
Description
Keywords
Nonlinear eigenvalue problem, Steklov problem, p-Laplacian, Nonlinear boundary conditions, Indefinite weight
Citation
Torné, O. (2005). Steklov problem with an indefinite weight for the p-Laplacian. <i>Electronic Journal of Differential Equations, 2005</i>(87), pp. 1-9.
Rights
Attribution 4.0 International