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A Priori Bounds and the Existence of Positive Solutions for Weighted Fractional Systems

Authors
  • Wang, Pengyan1
  • Niu, Pengcheng1
  • 1 Northwestern Polytechnical University, Xi’an, 710129, China , Xi’an (China)
Type
Published Article
Journal
Acta Mathematica Scientia
Publisher
Springer-Verlag
Publication Date
Jun 29, 2021
Volume
41
Issue
5
Pages
1547–1568
Identifiers
DOI: 10.1007/s10473-021-0509-2
Source
Springer Nature
Keywords
Disciplines
  • Article
License
Yellow

Abstract

In this paper, we prove the existence of positive solutions to the following weighted fractional system involving distinct weighted fractional Laplacians with gradient terms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ {\matrix{{( - \Delta )_{{a_1}}^{{\alpha \over 2}}{u_1}(x) = u_1^{{q_{11}}}(x) + u_2^{{q_{12}}}(x) + {h_1}(x,{u_1}(x),{u_2}(x),\nabla {u_1}(x),\nabla {u_2}(x)),\,\,\,\,\,\,x \in \Omega ,} \hfill \cr {( - \Delta )_{{a_2}}^{{\beta \over 2}}{u_2}(x) = u_1^{{q_{21}}}(x) + u_2^{{q_{22}}}(x) + {h_2}(x,{u_1}(x),{u_2}(x),\nabla {u_1}(x),\nabla {u_2}(x)),\,\,\,\,\,\,x \in \Omega ,} \hfill \cr {{u_1}(x) = 0,\,\,\,{u_2}(x) = 0,\,\,\,\,\,\,x \in {\mathbb{R}}{^n}\backslash \Omega .} \hfill \cr } } \right.$$\end{document} Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$( - \Delta )_{{a_1}}^{{\alpha \over 2}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$( - \Delta )_{{a_2}}^{{\beta \over 2}}$$\end{document} denote weighted fractional Laplacians and Ω ⊂ ℝn is a C2 bounded domain. It is shown that under some assumptions on hi(i = 1, 2), the problem admits at least one positive solution (u1(x), u2(x)). We first obtain the a priori bounds of solutions to the system by using the direct blow-up method of Chen, Li and Li. Then the proof of existence is based on a topological degree theory.

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