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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.

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
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