Bifurcation analysis of the Hardy-Sobolev equation

Abstract In this paper, we prove existence of multiple non-radial solutions to the Hardy-Sobolev equation { − Δ u − γ | x | 2 u = 1 | x | s | u | p s − 2 u  in  R N ∖ { 0 } , u ≥ 0 , where N ≥ 3 , s ∈ [ 0 , 2 ) , p s = 2 ( N − s ) N − 2 and γ ∈ ( − ∞ , ( 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.