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2024 Existence of Solutions for Fractional $(p,q)$-Laplacian Problems Involving Critical Hardy–Sobolev Nonlinearities
Xuehui Cui, Yang Yang
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Taiwanese J. Math. Advance Publication 1-21 (2024). DOI: 10.11650/tjm/240402


This paper is devoted to studying a class of fractional $(p,q)$-Laplacian problems with subcritical and critical Hardy potentials: \[ \begin{cases} (-\Delta)^{s_{1}}_{p} u + \nu (-\Delta)^{s_{2}}_{q} u = \lambda \frac{|u|^{r-2} u}{|x|^{a}} + \frac{|u|^{p^{*}_{s_{1}}(b)-2} u}{|x|^{b}} &\textrm{in $\Omega$}, \\ u = 0 &\textrm{in $\mathbb{R}^{N} \setminus \Omega$}, \end{cases} \] where $\Omega \subset \mathbb{R}^{N}$ is a smooth and bounded domain, and $p_{s_{1}}^{*}(b) = \frac{(N-b)p}{N-ps_{1}}$ denotes the fractional critical Hardy–Sobolev exponent. More precisely, when $\nu = 1$ and $\nu \gt 0$ is sufficiently small, using some asymptotic estimates and the Mountain Pass Theorem, we establish the existence results for the above fractional elliptic equation under some suitable hypotheses, respectively, which are gained over a wider range of parameters.

Funding Statement

This project was supported by Natural Science Foundation of China (Nos. 11501252 and 11571176).


The authors would like to sincerely thank the anonymous referees for their valuable comments that helped to improve the manuscript.


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Xuehui Cui. Yang Yang. "Existence of Solutions for Fractional $(p,q)$-Laplacian Problems Involving Critical Hardy–Sobolev Nonlinearities." Taiwanese J. Math. Advance Publication 1 - 21, 2024.


Published: 2024
First available in Project Euclid: 22 April 2024

Digital Object Identifier: 10.11650/tjm/240402

Primary: 35A15 , 35B33 , 35J60 , 35R11

Keywords: Critical nonlinearity , fractional $(p,q)$-Laplacian , subcritical and critical Hardy exponents , variational methods

Rights: Copyright © 2024 The Mathematical Society of the Republic of China

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