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Some remarks on a nonhomogeneous eigenvalue problem related to generalized trigonometric functions. (English) Zbl 1461.34045

The quasi-linear eigenvalue problem for the \(p\)-Laplacian operator \[\left (|u'|^{p-2}u'\right)' +\frac{ q (p - 1)}p (\lambda - \omega (t)) |u|^{q-2} u = 0, \ t \in (0, 1)\] for \(p,q>1\) and \(\omega \in C^1(0,1)\) subject to the boundary conditions \(u' (0) = u'(1) = 0\) and \(u(0) = \alpha \), \(\alpha >0\), is considered. It was shown by P. Drábek and R. Manásevich [Differ. Integral Equ. 12, No. 6, 773–788 (1999; Zbl 1015.34071)] that the eigenfunctions and eigenvalues in case \(\omega =0\) can be expressed in terms of the generalized sine functions \(\sin_{p,q}\), which are periodic and whose inverse on a suitable quarter-period is defined by \(\arcsin_{p,q}(\sigma )=\int_0^\sigma (1-\sigma ^q)^{-\frac 1p}ds\).
The paper has four main results. Theorem 1.2 states an asymptotic expansion of the eigenvalues in terms of the zeros of \(\sin_{p,q}\). For \(p>\max\{2,4q(4q-\pi^2+8)^{-1}\}\), Theorem 1.3 states a Schauder basis result in \(L^r(0,1)\), \(r>1\), for the eigenfunctions, where finitely many eigenfunctions are replaced by eigenfunctions of the problem with \(\omega =0\). In Theorem 1.4 it is shown that \(\omega (t)-\int_0^1\omega (\tau )\,d\tau \) can be recovered as a limit of functions involving nodal points of the eigenfunctions. Theorem 1.5 gives the generalization of Ambarzumyan’s theorem (the case \(p=q=2\)) to the corresponding inverse problem for arbitrary \(p,q>1\).

MSC:

34B09 Boundary eigenvalue problems for ordinary differential equations
34A55 Inverse problems involving ordinary differential equations
34B24 Sturm-Liouville theory
34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators

Citations:

Zbl 1015.34071
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