Existence and multiplicity of positive solutions for a critical fractional Laplacian equation with singular nonlinearity

IF 2.5 2区数学 Q1 MATHEMATICS

Fractional Calculus and Applied Analysis Pub Date : 2024-02-16 DOI:10.1007/s13540-024-00242-y

引用次数: 0

Abstract

In this paper, we consider the following problem $$\begin{aligned} {\left\{ \begin{array}{ll} (-\varDelta )^{s} u=g(x) u^{2_{s}^{*}-1}+\lambda u^{-\gamma }, &{} \text { in } \varOmega , \\ u>0, \text { in } \varOmega , \quad u=0, &{} \text { on } \partial \varOmega , \end{array}\right. } \end{aligned}$$ where $\varOmega \subset {\mathbb {R}}^{N}(N > 2s)$ is a smooth bounded domain, $s\in (0,1)$ , $\lambda $ is a positive constant, $0<\gamma <1$ , $2_{s}^{*}=\frac{2 N}{N-2s}$ and $(-\varDelta )^{s} $ is the spectral fractional Laplacian. Based upon the Nehari manifold and using variational method we relate the number of positive solutions to the global maximum of the coefficient of the critical nonlinearity g.

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具有奇异非线性的临界分数拉普拉斯方程正解的存在性和多重性

Abstract In this paper, we consider following problem $$\begin{aligned} {\left\{ \begin{array}{ll} (-\varDelta )^{s} u=g(x) u^{2_{s}^{*}-1}+\lambda u^{-\gamma }, &{}\text { in }\varOmega , \ u>0, \text { in }\varOmega , \quad u=0, \amp;{}\text { on }\partial \varOmega , \end{array}\right.}\end{aligned}$$ 其中 $\varOmega \subset {\mathbb {R}}^{N}(N > 2s)$ 是一个光滑的有界域, $s\in (0,1)$(\lambda)是一个正常数，$0<\gamma<1$，$2_{s}^{*}=\frac{2 N}{N-2s}$ 和$(-\varDelta )^{s}$是谱分数拉普拉奇。基于 Nehari 流形并使用变分法，我们将正解的数量与临界非线性系数 g 的全局最大值联系起来。

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

Fractional Calculus and Applied Analysis MATHEMATICS, APPLIED-MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

CiteScore

4.70

自引率

16.70%

发文量

101

期刊介绍： Fractional Calculus and Applied Analysis (FCAA, abbreviated in the World databases as Fract. Calc. Appl. Anal. or FRACT CALC APPL ANAL) is a specialized international journal for theory and applications of an important branch of Mathematical Analysis (Calculus) where differentiations and integrations can be of arbitrary non-integer order. The high standards of its contents are guaranteed by the prominent members of Editorial Board and the expertise of invited external reviewers, and proven by the recently achieved high values of impact factor (JIF) and impact rang (SJR), launching the journal to top places of the ranking lists of Thomson Reuters and Scopus.