## 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

Zbl 1015.34071
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