Anane, A.; Chakrone, O.; Moussa, M. Spectrum of one dimensional \(p\)-Laplacian operator with indefinite weight. (English) Zbl 1022.35033 Electron. J. Qual. Theory Differ. Equ. 2002, Paper No. 17, 11 p. (2002). Summary: This paper is concerned with the nonlinear boundary eigenvalue problem \[ -(|u'|^{p-2}u')'=\lambda m|u|^{p-2}u,\qquad u \in I=]a,b[,\quad u(a)=u(b)=0, \] where \(p>1\), \(\lambda\) is a real parameter, \(m\) is an indefinite weight, and \(a\), \(b\) are real numbers. We prove there exists a unique sequence of eigenvalues for this problem. Each eigenvalue is simple and verifies the strict monotonicity property with respect to the weight \(m\) and the domain \(I\), the \(k\)-th eigenfunction, corresponding to the \(k\)-th eigenvalue, has exactly \(k-1\) zeros in \((a,b)\). At the end, we give a simple variational formulation of eigenvalues. Cited in 26 Documents MSC: 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 35J70 Degenerate elliptic equations Keywords:\(p\)-Laplacian spectrum; simplicity; isolation; strict monotonicity property; zeros of eigenfunctions; variational formulation PDFBibTeX XMLCite \textit{A. Anane} et al., Electron. J. Qual. Theory Differ. Equ. 2002, Paper No. 17, 11 p. (2002; Zbl 1022.35033) Full Text: EuDML EMIS