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On the first eigencurve of the \(p\)-Laplacian. (English) Zbl 1142.35537

Benkirane, Abdelmoujib (ed.) et al., Partial differential equations. Proceedings of the international conference, Fez, Morocco. New York, NY: Marcel Dekker (ISBN 0-8247-0780-X/pbk). Lect. Notes Pure Appl. Math. 229, 195-205 (2002).
The present paper is devoted to the following problem \[ \begin{gathered} -\Delta_p u=\lambda m(x)|u|^{p-2} u+ \mu||^{p-2} u\quad\text{in }\Omega,\\ u\in W^{1,p}_0(\Omega),\end{gathered}\tag{1} \] where \(\Delta_p\) is the \(p\)-Laplacian, \(p> 1\) and \(m\in L^\infty(\Omega)\) with \(m\not\equiv 0\) which can be change the sign, \(\lambda\in\mathbb{R}\), \(\mu\in\mathbb{R}\), and \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(N\geq 1\) not necessarily regular. The authors show that for any \(\lambda\in\mathbb{R}\) the first eigenvalue \(\mu_1(\lambda)\) of (1) is simple and isolated in any bounded domain \(\Omega\). Moreover, they prove some results about variations of the weight. The technique of the authors based on the variational principle.
For the entire collection see [Zbl 0986.00055].

MSC:

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J25 Boundary value problems for second-order elliptic equations
35J60 Nonlinear elliptic equations
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