Eigenvalue problems with p-Laplacian operators

In this article, we study eigenvalue problems with the p-Laplacian operator: $$-(|y'|^{p-2}y')'= (p-1)(\lambda\rho(x)-q(x))|y|^{p-2}y \quad \text{on } (0,\pi_{p}),$$ where p>1 and $\pi_{p}\equiv 2\pi/(p\sin(\pi/p))$. We show that if $\rho \equiv 1$ and q is single-well with transition point $a=\pi_{p}/2$, then the second Neumann eigenvalue is greater than or equal to the first Dirichlet eigenvalue; the equality holds if and only if q is constant. The same result also holds for p-Laplacian problem with single-barrier $\rho$ and $q \equiv 0$. Applying these results, we extend and improve a result by [24] by using finitely many eigenvalues and by generalizing the string equation to p-Laplacian problem. Moreover, our results also extend a result of Huang [14] on the estimate of the first instability interval for Hill equation to single-well function q

Nodal properties for p-Laplacian systems

We consider a system of differential equations involving the p-Laplacian. We prove the existence of oscillatory solutions with prescribed numbers of zeros, and show that the solutions satisfy the Dirichlet boundary conditions when the large parameters in the equations are suitable chosen. Our main tool in this work is a Prufer-type substitution

The Partial Inverse Spectral and Nodal Problems for Sturm–Liouville Operators on a Star-Shaped Graph

We firstly prove the Horváth-type theorem for Sturm–Liouville operators on a star-shaped graph and then solve a new partial inverse nodal problem for this operator. We give some algorithms to recover this operator from a dense nodal subset and prove uniqueness theorems from paired-dense nodal subsets in interior subintervals having a central vertex. In particular, we obtain some uniqueness theorems by replacing the information of nodal data on some fixed edge with part of the eigenvalues under some conditions

Distribution of the Prufer angle in p-Laplacian eigenvalue problems

The Prufer angle is an effective tool for studying Sturm-Liouville problems and p-Laplacian eigenvalue problems. In this article, we show that for the p-Laplacian eigenvalue problem, when x is irrational in (0,1), a sequence of modified Prufer angles (after modulo $\pi_p$) is equidistributed in $(0,\pi_p)$. As a function of x, $\psi_n$ is also asymptotic to the uniform distribution on $(0,\pi_p)$

The Partial Inverse Spectral and Nodal Problems for Sturm–Liouville Operators on a Star-Shaped Graph

We firstly prove the Horváth-type theorem for Sturm–Liouville operators on a star-shaped graph and then solve a new partial inverse nodal problem for this operator. We give some algorithms to recover this operator from a dense nodal subset and prove uniqueness theorems from paired-dense nodal subsets in interior subintervals having a central vertex. In particular, we obtain some uniqueness theorems by replacing the information of nodal data on some fixed edge with part of the eigenvalues under some conditions