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Article Dans Une Revue Topological Methods in Nonlinear Analysis Année : 2018

On multiplicity of eigenvalues and symmetry of eigenfunctions of the p-Laplacian

Résumé

We investigate multiplicity and symmetry properties of higher eigenvalues and eigenfunctions of the p-Laplacian under homogeneous Dirichlet boundary conditions on certain symmetric domains Ω ⊂ R^N. By means of topological arguments, we show how symmetries of Ω help to construct subsets of W_0^(1,p)(Ω) with suitably high Krasnosel'ski˘ ı genus. In particular, if Ω is a ball B ⊂ R^N , we obtain the following chain of inequalities: λ_2(p; B) ≤ · · · ≤ λ_(N+1)(p; B) ≤ λ_eq(p; B). Here λ_i(p; B) are variational eigenvalues of the p-Laplacian on B, and λ_eq(p; B) is the eigenvalue which has an associated eigenfunction whose nodal set is an equatorial section of B. If λ_2(p; B) = λ(p; B), as it holds true for p = 2, the result implies that the multiplicity of the second eigenvalue is at least N. In the case N = 2, we can deduce that any third eigenfunction of the p-Laplacian on a disc is nonradial. The case of other symmetric domains and the limit cases p = 1, p = ∞ are also considered.
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Dates et versions

hal-01505548 , version 1 (11-04-2017)

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Benjamin Audoux, Vladimir Bobkov, Enea Parini. On multiplicity of eigenvalues and symmetry of eigenfunctions of the p-Laplacian. Topological Methods in Nonlinear Analysis, 2018, 51 (2), pp.565-582. ⟨10.12775/TMNA.2017.055⟩. ⟨hal-01505548⟩
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