Non-existence for a semi-linear fractional system with Sobolev exponents via direct method of moving spheres

Abstract

The aim of this article is to consider the semi-linear fractional system with Sobolev exponents q = frac{{n + alpha}}{{n - beta}} and p = frac{{n + beta}}{{n - alpha}} (alpha nebeta): {(−Δ)α/2u(x)=k(x)vq(x)+f(v(x)),(−Δ)β/2v(x)=j(x)up(x)+g(u(x)),\\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}$$\\left \\{ \\textstyle\\begin{array}{l} {( - \\Delta)^{\\alpha/2}}u(x) = k(x){v^{q}}(x) + f(v(x)),\\\\ {( - \\Delta)^{\\beta/2}}v(x) = j(x){u^{p}}(x) + g(u(x)), \\end{array}\\displaystyle \\right . $$\\end{document} where 0 < alpha, beta < 2. We first establish two maximum principles for narrow regions in the ball and out of the ball by the iteration technique, respectively. Based on these principles, we use the direct method of moving spheres to prove the non-existence of positive solutions to the above system in the whole space and bounded star-shaped domain. As a consequence, the monotonic decreasing properties of W(x) = { vert x vert ^{frac{{n - alpha}}{2}}}u(x) and {W_{1}}(x) = { vert x vert ^{frac{{n - beta}}{2}}}v(x) along the radial direction in the whole space are obtained.