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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

${{\Delta }}_{p}u\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}\left(q-\lambda r\right){|u|}^{p-1}\text{sgn}\phantom{\rule{4pt}{0ex}}u\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega },$

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 ${{\Delta }}_{p}$ is the $p$-Laplacian with $1, ${\Omega }$ is a smooth domain in ${ℝ}^{N}$, and $\phantom{\rule{4pt}{0ex}}q,r\in {C}^{1}\left(\overline{{\Omega }}\right)$ 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.

##### MSC:
 35P30 Nonlinear eigenvalue problems for PD operators; nonlinear spectral theory 47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
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