On subhomogeneous indefinite p-Laplace equations in the supercritical spectral interval
On subhomogeneous indefinite p-Laplace equations in the supercritical spectral interval
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超临界谱区间内次齐次不定p-拉普拉斯方程
DOI：
10.1007/s00526-022-02322-4
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发表时间：
2023
影响因子：
2.1
通讯作者：
Vladimir Bobkov and Mieko Tanaka
中科院分区：
文献类型：
--
作者：
Vladimir Bobkov and Mieko Tanaka
研究了有界域上方程的零Dirichlet问题解的存在性、多解性和某些定性性质，其中，和是变号权函数。我们的主要兴趣是基态和非负解，它们是正的，当参数位于临界值的邻域内时，如果参数位于临界值的邻域内，则参数位于临界值的邻域内。当参数位于临界值的邻域内时，参数为：=\inf\Left\{\int_\ommega|\nabla|^p\，dx/int_\ommega|amsbsy}\usepackage{mathsfs}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\Begin{Document}$\lambda^*：=\inf\Left\{\int_\Omega|\nabla|^p\，dx/int_\Omega|u|^pDx：u\in W_0^{1，p}(\Omega){\setminus}\{0\}，\\int_\Omega a|u|^q\，dx\ge 0\，\right\}$\end{文档}。在主要结果中，我们证明了如果和或充分小，则这样的解确实存在于的邻域内。这是文[1]中Dirichletp-Laplace算子的第一个本征函数。这种存在现象具有纯粹的次齐次和非线性性质，因为无论是在超齐次情况下，还是在次线性情况下，对任何一种情况都不存在。此外，我们证明了如果AND足够小，则在的一个邻域内存在三个非零非负解，其中两个非负解是严格正的。
We study the existence, multiplicity, and certain qualitative properties of solutions to the zero Dirichlet problem for the equationin a bounded domain, where,, andais a sign-changing weight function. Our primary interest concerns ground states and nonnegative solutions which are positive in, when the parameterlies in a neighborhood of the critical value \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^* := \inf \left\{ \int _\Omega |\nabla u|^p \, dx/\int _\Omega |u|^p \, dx: u\in W_0^{1,p}(\Omega ) {\setminus } \{0\},\ \int _\Omega a|u|^q\,dx \ge 0\,\right\} $$\end{document}. Among main results, we show that ifand eitheroris sufficiently small, then such solutions do exist in arightneighborhood of. Hereis the first eigenfunction of the Dirichletp-Laplacian in. This existence phenomenon is of a purely subhomogeneous and nonlinear nature, since either in the superhomogeneous caseor in the sublinear casethe nonexistence takes place for any. Moreover, we prove that ifandis sufficiently small, then there existthreenonzero nonnegative solutions in aleftneighborhood of, two of which are strictly positive in.