Cuesta, Mabel Eigenvalue problems for the \(p\)-Laplacian with indefinite weights. (English) Zbl 0964.35110 Electron. J. Differ. Equ. 2001, Paper No. 33, 9 p. (2001). Summary: We consider the eigenvalue problem \(-\Delta_p u=\lambda V(x) |u|^{p-2} u\), \(u\in W_0^{1,p} (\Omega)\) where \(p>1\), \(\Delta_p\) is the p-Laplacian operator, \(\lambda >0\), \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) and \(V\) is a given function in \(L^s (\Omega)\) (\(s\) depending on \(p\) and \(N\)). The weight function \(V\) may change sign and has a nontrivial positive part. We prove that the least positive eigenvalue is simple, isolated in the spectrum and it is the unique eigenvalue associated to a nonnegative eigenfunction. Furthermore, we prove the strict monotonicity of the least positive eigenvalue with respect to the domain and the weight. Cited in 57 Documents MSC: 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 35J70 Degenerate elliptic equations 35J20 Variational methods for second-order elliptic equations 35P05 General topics in linear spectral theory for PDEs Keywords:nonlinear eigenvalue roblem; \(p\)-Laplacian; indefinite weight PDFBibTeX XMLCite \textit{M. Cuesta}, Electron. J. Differ. Equ. 2001, Paper No. 33, 9 p. (2001; Zbl 0964.35110) Full Text: EuDML EMIS