On the generalization of the Courant nodal domain theorem (Q1604633)

This paper concerns with the nodal properties of the eigenfunctions (Fučik eigenfunctions) of \(-\Delta_p\), where \(p>1\), \(\Delta_pu:= \nabla\cdot(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplacian. More precisely, the authors study the properties of the nonlinear eigenvalue problem \[ -\Delta_pu= \lambda|u|^{p-2}u \quad\text{in }\Omega, \qquad u=0 \quad\text{on }\partial\Omega, \] and its more general version \[ -\Delta_pu= \alpha|u|^{p-2}u^+-\beta|u|^{p-2}u^- \quad\text{in }\Omega, \qquad u=0 \quad\text{on }\partial\Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb R^N\) with smooth boundary \(\partial\Omega\), and \(\alpha,\beta,\lambda\) are real spectral parameters. The authors prove that, if \(u_{\lambda_n}\) is an eigenfunction associated with the \(n\)th variational eigenvalue, \(\lambda_n\), then \(u_{\lambda_n}\) has at most \(2n-2\) nodal domains. Moreover, if \(u_{\lambda_n}\) has \(n+k\) nodal domains then there is another eigenfunction with at most \(n-k\) nodal domains.