Spectrum of one-dimensional \(p\)-Laplacian with an indefinite integrable weight
For \(1 < p < \infty\), the authors study the spectrum of the one-imensional \(p\)-Laplacian \[ (\varphi_p (x'))' + \mu \omega (t) \varphi (x) = 0\text{ on }[0,1] \] with Dirichlet boundary conditions \(x(0) = x(1) = 0\), where \(\varphi_p (x) := |x|^{p-2}x\) and \(\omega\) is a (generally) indefinite weight in \(L^\gamma([0,1], \mathbb{R})\) \((=: \mathcal{L}^\gamma)\) with \(1 \leq \gamma \leq \infty\). If \(\omega\) is positive, then it is well known that the spectrum consists of an increasing sequence of positive eigenvalues accumulating at \(\infty\). If \(\omega\) allows a sign change but is still strictly positive on a set of non-zero measure, then it is shown by means of Prüfer transformations that this property remains true for the non-negative part of the spectrum. The set \(W_+^\gamma\) of all such weights is an open subset of \(\mathcal{L}^\gamma\) with respect to the usual norm \(\|\cdot\|_\gamma\). The two main results of the paper deal with the dependence of the \(m\)th positive eigenvalue \(\mu_m^+(\omega) \; (m \in \mathbb{N})\) on the weight, i.e., with the mapping \(\omega \rightarrow \mu_m^+(\omega)\): 1. As a mapping \(W_+^\gamma \rightarrow (0,\infty)\), it is continuously differentiable with respect to \(\|\cdot\|_\gamma\) and the differential can be expressed in terms of eigenfunctions. 2. With respect to the weak topology in \(\mathcal{L}^\gamma\), the same mapping is continuous. Moreover, similar (but more complicated) results are also obtained for Neumann (instead of Dirichlet) boundary conditions. Finally, the results are applied to extremal problems of eigenvalues, i.e., to estimates of \(\inf \mu_m^+(\omega)\) and \(\sup \mu_m^+(\omega)\), where \(\omega\) varies over certain subsets of \(\mathcal{L}^\gamma\).
scientific article; zbMATH DE number 5728337
scientific article