Solutions to the nonlinear Schrodinger systems involving the fractional Laplacian.

ABSTRACT: In this paper, we consider the following nonlinear Schrödinger system involving the fractional Laplacian operator: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} (-\Delta)^{\frac{\alpha}{2}}u+au=f(v), \\ (-\Delta)^{\frac{\beta}{2}}v+bv=g(u), \end{cases}\displaystyle \quad \text{on } \Omega\subseteqq \mathbb{R}^{n}, $$\end{document}{(??)?2u+au=f(v),(??)?2v+bv=g(u),on ??Rn, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a,b\geq0$\end{document}a,b?0. When ? is the unit ball or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{n}$\end{document}Rn, we prove that the solutions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(u,v)$\end{document}(u,v) are radially symmetric and decreasing. When ? is the parabolic domain on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{n}$\end{document}Rn, we prove that the solutions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(u,v)$\end{document}(u,v) are increasing. Furthermore, if ? is the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{n}_{+}$\end{document}R+n, then we also derive the nonexistence of positive solutions to the system on the half-space. We assume that the nonlinear terms f, g and the solutions u, v satisfy some amenable conditions in different cases.

SUBMITTER: Qu M 

PROVIDER: S-EPMC6208611 | biostudies-other | 2018

REPOSITORIES: biostudies-other