On the Fučik spectrum of Hardy-Sobolev operator
The author deals with the Fučik spectrum of the Hardy-Sobolev operator by \(L_\mu\) on \(W^{1,p}_0(\Omega)\) defined as the set \(\Sigma_{p,\mu}\) of these \((\alpha,\beta)\in\mathbb{R}^2\) such that \[ \begin{gathered} L_\mu u:= -\Delta_pu- {\mu\over|x|^p}|u|^{p- 2}u= \alpha(u^+)^{p-1}- \beta(u^-)^{p-1}\quad\text{in }\Omega,\\ u= 0\quad\text{on }\partial\Omega\end{gathered} \] has a nontrivial solution \(u\in W^{1,p}_0(\Omega)\). Here \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) containing \(0\), \(N\geq 2\), \(1< p< n\), \(\Delta_p\) is the \(p\)-Laplacian, \(\mu<({n-p\over p})^p\). Using comparison theorem the author shows that the first eigenvalue \(\mu_1\) is simple and isolated in the spectrum \(L_\mu\). Moreover, the author constructs a nontrivial curve \(C\) in \(\Sigma_{p,\mu}\), that is so-called the ``first curve'' and proves that the two trivial lines \(\mu_1\times\mathbb{R}\) and \(\mathbb{R}\times \mu_1\) are isolated in \(\Sigma_{p,\mu}\).
scientific article; zbMATH DE number 1836638