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On the second eigenvalue of the \(p\)-Laplacian. (English) Zbl 0854.35081

Benkirane, A. (ed.) et al., Nonlinear partial differential equations. Based on the international conference on nonlinear analysis, Fés, Morocco, May 9-14, 1994. Harlow: Longman. Pitman Res. Notes Math. Ser. 343, 1-9 (1996).
The eigenvalue problem \[ - \text{div}(|\nabla u|^{p- 2} \nabla u)= \lambda m(x) |u|^{p- 2} u\quad \text{in } \Omega,\quad u= 0\quad \text{on } \partial\Omega \] is studied where \(1< p< \infty\), \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(m\in L^\infty\) is a weight function, \(\text{meas}\{x\in \Omega; m(x)> 0\}> 0\). An estimate of the number of components of the nodal set \(\{x\in \Omega; u(x)\neq 0\}\) of the \(n\)th eigenfunction is given which is an analogy of the results known for the case \(p= 2\). A variational characterization of the second eigenvalue is proved on the basis of this estimate. Further, a monotone dependence of the second eigenvalue on the weight \(m\) is proved.
For the entire collection see [Zbl 0837.00016].
Reviewer: M.Kučera (Praha)

MSC:

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
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