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Zbl 1136.35061
Binding, Paul A.; Rynne, Bryan P.
Variational and non-variational eigenvalues of the $p$-Laplacian.
(English)
[J] J. Differ. Equations 244, No. 1, 24-39 (2008). ISSN 0022-0396

This paper is concerned with nonlinear eigenvalue problems $$\Delta_p u \ = \ (q-\lambda r) \vert u\vert ^{p-1} \text{sgn}\ u \quad\text{in }\Omega,$$ for which not all eigenvalues are of variational type in the sense of Ljusternik-Schnirelman (1934; Zbl 0011.02803); see e.g. {\it H. Amann} [Math. Ann. 199, 55--71 (1972; Zbl 0233.47049)]. Here $\Delta_p$ is the $p$-Laplacian with $1<p\ne 2$, $\Omega$ is a smooth domain in $\Bbb R^N$, and $\ q,r\in C^1(\overline{\Omega})$ are coefficients with $r>0$ on $\overline{\Omega}$. 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.
[In-Sook Kim (München)]
MSC 2000:
*35P30 Nonlinear eigenvalue problems for PD operators
47J10 Nonlinear eigenvalue problems

Keywords: nonlinear eigenvalue problems; $p$-Laplacian; variational eigenvalues; Carathéodory eigenvalues

Citations: Zbl 0233.47049; Zbl 0011.02803

Cited in: Zbl 1195.47041

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