Eigenvalue problems for the \(p\)-Laplacian with indefinite weights. (English) Zbl 0964.35110

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.


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
Full Text: EuDML EMIS