Classification of nonnegative solutions to static Schr\xf6dinger\u2013Hartree\u2013Maxwell system involving the fractional Laplacian

Abstract

This paper is mainly concerned with the following semi-linear system involving the fractional Laplacian: {(−Δ)α2u(x)=(1|⋅|σ∗vp1)vp2(x),x∈Rn,(−Δ)α2v(x)=(1|⋅|σ∗uq1)uq2(x),x∈Rn,u(x)≥0,v(x)≥0,x∈Rn,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} (-\\Delta )^{\\frac{\\alpha }{2}}u(x)= (\\frac{1}{ \\vert \\cdot \\vert ^{\\sigma }} \\ast v^{p_{1}} )v^{p_{2}}(x), \\quad x\\in \\mathbb{R}^{n}, \\\\ (-\\Delta )^{\\frac{\\alpha }{2}}v(x)= (\\frac{1}{ \\vert \\cdot \\vert ^{\\sigma }} \\ast u^{q_{1}} )u^{q_{2}}(x), \\quad x\\in \\mathbb{R}^{n}, \\\\ u(x)\\geq 0,\\quad\\quad v(x)\\geq 0, \\quad x\\in \\mathbb{R}^{n}, \\end{cases} $$\\end{document} where 0<alpha leq 2, ngeq 2, 0<sigma <n, and 0< p_{1}, q_{1}leq frac{2n-sigma }{n-alpha }, 0< p_{2}, q_{2}leq frac{n+alpha -sigma }{n-alpha }. Applying a variant (for nonlocal nonlinearity) of the direct method of moving spheres for fractional Laplacians, which was developed by W. Chen, Y. Li, and R. Zhang (J. Funct. Anal. 272(10):4131–4157, 2017), we derive the explicit forms for positive solution (u,v) in the critical case and nonexistence of positive solutions in the subcritical cases.