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Variational and non-variational eigenvalues of the p-Laplacian. (English) Zbl 1136.35061

This paper is concerned with nonlinear eigenvalue problems

Δ p u=(q-λr)|u| p-1 sgnuinΩ,

for which not all eigenvalues are of variational type in the sense of Ljusternik-Schnirelman (1934; Zbl 0011.02803); see e.g. H. Amann [Math. Ann. 199, 55–71 (1972; Zbl 0233.47049)]. Here Δ p is the p-Laplacian with 1<p2, Ω is a smooth domain in N , and q,rC 1 (Ω ¯) are coefficients with r>0 on Ω ¯.

Some examples are given for ordinary differential equations with periodic boundary conditions and partial differential equations with Neumann boundary conditions, in the case of non-constant coefficients. Moreover, for the periodic problem, the variational eigenvalues are characterized via an extremal property within the set of Carathéodory eigenvalues.

35P30Nonlinear eigenvalue problems for PD operators; nonlinear spectral theory
47J10Nonlinear spectral theory, nonlinear eigenvalue problems
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