The dual eigenvalue problems for \(p\)-Laplacian (Q2439818)

Consider the one-dimensional \(p\)-Laplacian defined by \[ \begin{aligned} &-(y^{\prime(p-1)})'=(p-1)(\lambda\rho(x)-q(x)) y^{( p-1)}, \text{where\;}0\leq x\leq \widehat{\pi},\\ &y(0)=y(\widehat{\pi})=0.\end{aligned} \] Here \(y^{(p-1)}=| y| ^{(p-2)}y\), \(\widehat{\pi}=2\pi/( p\sin( \pi/p)),\rho(x)>0\) a.e., \(q,\rho\in L(0,\widehat{\pi})\) and \(p>1.\) The paper shows that, when \(q\) is a single well potential with a transition at \(x=\widehat{\pi}/2\), we have \(| \lambda_{2}-\lambda_{1}| \geq2^{p}-1\). The second result considers the case when \(q=0\) and shows that when the weight \(\rho\) is a single barrier function, then the ratio \(\mu_{2}/\mu_{1}\geq2^{p}\). The authors show that these results are best possible, and use Sturmian type theorems to compare eigenfunctions and their zeros.