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On multiplicity of eigenvalues and symmetry of eigenfunctions of the \(p\)-Laplacian. (English) Zbl 1410.35050
Authors’ abstract: We investigate multiplicity and symmetry properties of higher eigenvalues and eigenfunctions of the \(p\)-Laplacian under homogeneous Dirichlet boundary conditions on certain symmetric domains \(\Omega \subset \mathbb{R}^N.\) By means of topological arguments, we show how symmetries of \(\Omega\) help to construct subsets of \(W_0^{1,p}(\Omega)\) with suitably high Krasnosel’skiĭ genus. In particular, if \(\Omega\) is a ball \(B \subset \mathbb{R}^N,\) we obtain the following chain of inequalities: \[ \lambda_2(p;B) \leq \dots \leq \lambda_{N+1}(p;B) \leq \lambda_\ominus(p;B). \] Here \(\lambda_i(p;B)\) are variational eigenvalues of the \(p\)-Laplacian on \(B\), and \(\lambda_\ominus(p;B)\) is the eigenvalue which has an associated eigenfunction whose nodal set is an equatorial section of \(B\). If \(\lambda_2(p;B) = \lambda_\ominus(p;B)\), 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 \(p=1\), \(p=\infty\) are also considered.
MSC:
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35B06 Symmetries, invariants, etc. in context of PDEs
35A15 Variational methods applied to PDEs
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