论文标题
lyapunov型的不平等现象，即一维$ p $ -laplacian的Sturm-Liouville问题
Lyapunov-type inequalities for a Sturm-Liouville problem of the one-dimensional $p$-Laplacian
论文作者
论文摘要
本文考虑了Sturm-Liouville问题的特征问题，包括$ P $ -LAPLACIAN \ BEGIN {ALIGN*} \ BEGIN {Case} \ left（\ Vert U'\ Vert u'\ Vert u'\ vert^{p-2 {p-2} ^{p-2} u = 0，\，\，x \ in（0，π_{p}），\\ u（0）= u（π_{p}）= 0，\ end {cases} \ end {case} \ end {align {align {align {align {align {align {align*} $ 1 <p <p <p <p <p <\ p <\ fifty $ fifty $ fifty $，$π_ $π_{p} =2π/\ left（p \ sin（π/p）\ right）$，$ r \ in c [0，π_{p}] $和$λ<p-1 $。提出了尖锐的lyapunov型不平等，这是上述问题的非平凡溶液的必要条件。结果是通过分析与尖锐的Sobolev嵌入以及广义三角和双曲线功能有关的分析。
This article considers the eigenvalue problem for the Sturm-Liouville problem including $p$-Laplacian \begin{align*} \begin{cases} \left(\vert u'\vert^{p-2}u'\right)'+\left(λ+r(x)\right)\vert u\vert ^{p-2}u=0,\,\, x\in (0,π_{p}),\\ u(0)=u(π_{p})=0, \end{cases} \end{align*} where $1<p<\infty$, $π_{p}$ is the generalized $π$ given by $π_{p}=2π/\left(p\sin(π/p)\right)$, $r\in C[0,π_{p}]$ and $λ<p-1$. Sharp Lyapunov-type inequalities, which are necessary conditions for the existence of nontrivial solutions of the above problem are presented. Results are obtained through the analysis of variational problem related to a sharp Sobolev embedding and generalized trigonometric and hyperbolic functions.
