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Zbl 1163.35449
Drábek, Pavel; Robinson, Stephen B.
On the generalization of the Courant nodal domain theorem.
(English)
[J] J. Differ. Equations 181, No. 1, 58-71 (2002). ISSN 0022-0396

This paper concerns with the nodal properties of the eigenfunctions (Fučik eigenfunctions) of $-\Delta_p$, where $p>1$, $\Delta_pu:= \nabla\cdot(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian. More precisely, the authors study the properties of the nonlinear eigenvalue problem $$-\Delta_pu= \lambda|u|^{p-2}u \quad\text{in }\Omega, \qquad u=0 \quad\text{on }\partial\Omega,$$ and its more general version $$-\Delta_pu= \alpha|u|^{p-2}u^+-\beta|u|^{p-2}u^- \quad\text{in }\Omega, \qquad u=0 \quad\text{on }\partial\Omega,$$ where $\Omega$ is a bounded domain in $\Bbb R^N$ with smooth boundary $\partial\Omega$, and $\alpha,\beta,\lambda$ are real spectral parameters. The authors prove that, if $u_{\lambda_n}$ is an eigenfunction associated with the $n$th variational eigenvalue, $\lambda_n$, then $u_{\lambda_n}$ has at most $2n-2$ nodal domains. Moreover, if $u_{\lambda_n}$ has $n+k$ nodal domains then there is another eigenfunction with at most $n-k$ nodal domains.
[Messoud A. Efendiev (Berlin)]
MSC 2000:
*35P30 Nonlinear eigenvalue problems for PD operators
47J10 Nonlinear eigenvalue problems
58E05 Abstract critical point theory

Keywords: nodal properties; eigenfunctions; Courant nodal domain; $p$-Laplacian

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