Bifurcation analysis of the Hardy-Sobolev equation

[摘要] In this paper, we prove existence of multiple non-radial solutions to the Hardy-Sobolev equation {-Delta u - gamma/vertical bar x vertical bar(2) u = 1/vertical bar x vertical bar(s) vertical bar u vertical bar(ps-2)u in R-N \ {0}, u >= 0, where N >= 3, s is an element of [0, 2), p(s) = 2(N-s)/N-2 and gamma is an element of (-infinity, (N-2)(2)/4). We extend results of E.N. Dancer, F. Gladiali, M. Grossi (2017) [12] where only the case s = 0 is considered. The results specially rely on a careful analysis of the kernel of the linearized operator. Moreover, thanks to monotonicity properties of the solutions, we separate two branches of non-radial solutions. (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-09-25

[关键词] Hardy-Sobolev inequality;Positive solutions;Morse index;Symmetry and monotonicity of solutions