Infinitely many new curves of the Fučík spectrum
This paper is devoted to some results on the Fučík spectrum for the Laplace operator, giving new information on its structure. These results show that if \(\Omega \) is a bounded connected domain of \(\mathbb R^N\), then the Fučík spectrum has infinitely many curves asymptotic to the lines \(\{\lambda_1\} \times\mathbb R\) and \(\mathbb R \times \{\lambda_1\}\), where \(\lambda_1\) denotes the first eigenvalue of the operator \(-\Delta\) in \(H^1_0(\Omega )\). If we set \(u^+=\max\{u, 0\}\) and \(u^-=\max\{-u, 0\}\), then the Fučík spectrum is defined as the set \(\Sigma\) of all pairs \((\alpha , \beta ) \in\mathbb R^2\) such that the Dirichlet problem \[ \begin{alignedat}{2} & \Delta u - \alpha u^- + \beta u^+ = 0 &\quad\text{in }& \Omega,\\ & u= 0, &\quad\text{on }&\partial\Omega,\end{alignedat} \] has nontrivial solutions. The main result of this paper may be stated as follows. \(\quad\) { Theorem 1.1} Let \(\Omega \) be a bounded connected domain of \(\mathbb R^N\) with \(N\geq 3\). Then, there exists a nondecreasing sequence \((b_k)_k\) of positive numbers with the following properties. For every positive integer \(k\) and for all \(\beta >b_k\), there exists \(\alpha_{k,\beta} >\lambda_1\) such that the pair \((\alpha_{k,\beta} , \beta )\) belongs to the Fučík spectrum \(\Sigma\). Moreover, for every positive integer \(k, \alpha_{k,\beta}\) depends continuously on \(\beta\) in \(]b_k, +\infty[, \alpha_{k,\beta} <\alpha_{k+1,\beta}\) for all \(\beta >b_{k+1}\) and \(\lim_{\beta\to\infty} \alpha_{k,\beta} =\lambda_1\). \(\quad\) To prove this theorem the authors use a variational method. For all \(\beta >0\), they consider the functional \[ f_\beta (u) =\int_\Omega [|Du|^2-\beta (u^+)^2]\,dx \] constrained on the set \(S=\{u \in H^1_0(\Omega ): \int_\Omega (u^-)^2\,dx=1\}\). For \(\beta >0\) large enough, the eigenfunction \(u\) is obtained as a constrained critical point for \(f_\beta\) on \(S\), while \(\alpha\) arises as the Lagrange multiplier with respect to the constraint \(S\).
scientific article; zbMATH DE number 6520567
scientific article