Critical p-Laplacian problems with a Hardy potential in a non-contractible domain

IF 0.9 3区数学 Q1 MATHEMATICS

Annali di Matematica Pura ed Applicata Pub Date : 2025-10-16 DOI:10.1007/s10231-025-01623-7

Diem Hang T. Le, Phuong Le

引用次数: 0

Abstract

We study the doubly critical quasilinear problem

$$ -\Delta _p u - \frac{\lambda }{|x|^p}|u|^{p-2}u = |u|^{p^*-2}u $$

in a bounded non-contractible domain $\Omega \subset \mathbb {R}^N$ containing the origin with zero Dirichlet boundary condition, where $1<p<N$, $0<\lambda <\left( \frac{N-p}{p}\right) ^p$ and $p^*$ is the critical Sobolev exponent. We show that the equation admits at least $\textrm{cat}(\Omega ) - 1$ positive solutions provided that $\lambda $ is small enough, where $\textrm{cat}(\Omega )$ denotes the Lusternik–Schnirelmann category of $\Omega $ in itself. Our result is new even in the case $p=2$, $N=3$.

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不可缩域上具有Hardy势的临界p-拉普拉斯问题

研究了含有零Dirichlet边界条件的原点的有界不可缩域$\Omega \subset \mathbb {R}^N$上的双临界拟线性问题$$ -\Delta _p u - \frac{\lambda }{|x|^p}|u|^{p-2}u = |u|^{p^*-2}u $$，其中$1<p<N$、$0<\lambda <\left( \frac{N-p}{p}\right) ^p$和$p^*$为临界Sobolev指数。我们证明，只要$\lambda $足够小，方程至少承认$\textrm{cat}(\Omega ) - 1$正解，其中$\textrm{cat}(\Omega )$表示$\Omega $本身的Lusternik-Schnirelmann范畴。我们的结果是新的，即使在$p=2$, $N=3$的情况下。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Annali di Matematica Pura ed Applicata 数学-数学

CiteScore

2.10

自引率

10.00%

发文量

审稿时长

>12 weeks

期刊介绍： This journal, the oldest scientific periodical in Italy, was originally edited by Barnaba Tortolini and Francesco Brioschi and has appeared since 1850. Nowadays it is managed by a nonprofit organization, the Fondazione Annali di Matematica Pura ed Applicata, c.o. Dipartimento di Matematica "U. Dini", viale Morgagni 67A, 50134 Firenze, Italy, e-mail annali@math.unifi.it). A board of Italian university professors governs the Fondazione and appoints the editors of the journal, whose responsibility it is to supervise the refereeing process. The names of governors and editors appear on the front page of each issue. Their addresses appear in the title pages of each issue.