Preprint
Inserted: 21 may 2026
Last Updated: 21 may 2026
Pages: 35
Year: 2026
Abstract:
We introduce a variational notion of essential spectrum for the Dirichlet $p-$Laplacian. We then extend the classical Persson Theorem to this nonlinear setting. This result provides a geometric characterization of the bottom of the essential spectrum, in terms of the sharp $L^p$ Poincar\'e constant ``at infinity''. We also show that in the case $p=2$ our construction of the essential spectrum is perfectly consistent with the classical theory. Finally, as an example, we compute the full spectrum of the Dirichlet $p-$Laplacian on a rectilinear strip: it is purely essential, with no embedded eigenvalues. The arguments of the proofs are elementary and new already for the linear case $p=2$.
Keywords: Nonlinear eigenvalue problems, $p-$Laplacian, essential spectrum, Persson's Theorem
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