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On some unexpected properties of radial and symmetric eigenvalues and eigenfunctions of the \(p\)-Laplacian on a disk. (English) Zbl 1366.35109
Authors’ abstract: We discuss several properties of eigenvalues and eigenfunctions of the \(p\)-Laplacian on a ball subject to zero Dirichlet boundary conditions. Among the main results, in two dimensions, we show the existence of nonradial eigenfunctions which correspond to the radial eigenvalues. Also, we prove the existence of eigenfunctions whose nodal sets take shapes that cannot occur in the linear case \(p=2\). Moreover, the limit behavior of some eigenvalues as \(p\to +1\) and \(p\to +\infty\) is studied.
Reviewer’s remarks: In the present paper, variational, radial and symmetric eigenvalues and corresponding eigenfunctions of the \(p\)-Laplacian on a ball are characterized in a precise manner, in comparison to the linear case \(p=2\). It may be regarded in part as a continuation of the very recent work of T. V. Anoop et al. [Proc. Am. Math. Soc. 144, No. 6, 2503–2512 (2016; Zbl 1386.35288)].

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35P15 Estimates of eigenvalues in context of PDEs
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
49R05 Variational methods for eigenvalues of operators
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