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Radial eigenfunctions for the game-theoretic \(p\)-Laplacian on a ball. (English) Zbl 1340.35242
Summary: The normalized or game-theoretic \(p\)-Laplacian operator given by \[ -\Delta_p^N u:=-\frac{1}{p}\left|\nabla u\right| ^{2-p}\Delta_p (u) \] for \(p \in (1,\infty)\) with \(\Delta_p u=\text{div}(\left|\nabla u\right|^{p-2} \nabla u)\) has no apparent variational structure. Showing the existence of a first (positive) eigenvalue of this fully nonlinear operator requires heavy machinery as in [I. Birindelli and F. Demengel, Z. Anal. Anwend. 29, No. 1, 77–90 (2010; Zbl 1185.35148)]. If it is restricted to the class of radial functions, however, the normalized \(p\)-Laplacian transforms into a linear Sturm-Liouville operator. We investigate radial eigenfunctions to this operator under homogeneous Dirichlet boundary conditions and come up with an explicit complete orthonormal system of Bessel functions in a suitably weighted \(L^2\)-space. This allows us to give a Fourier-series representation for radial solutions to the corresponding evolution equation \(u_t - \Delta_p^N u=0\).

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35B25 Singular perturbations in context of PDEs
35Q91 PDEs in connection with game theory, economics, social and behavioral sciences
35J60 Nonlinear elliptic equations
35J70 Degenerate elliptic equations