Bifurcation analysis of the Hardy-Sobolev equation

Abstract

In this paper, we prove existence of multiple non-radial solutions to the Hardy-Sobolev equation $$\{\begin{array}{ll}-\Delta u-\frac{\gamma }{|x{|}^2}u=\frac{1}{|x{|}^s}|u{|}^{p_s-2}u& \text{ in }{\mathbb{R}}^N\setminus \{0\},\\ u\ge 0,& \end{array}$$ where $N\ge 3$, $s\in [0,2)$, $p_s=\frac{2(N-s)}{N-2}$ and $\gamma \in (-\infty ,\frac{(N-2{)}^2}{4})$. We extend results of E.N. Dancer, F. Gladiali, M. Grossi, Proc. Roy. Soc. Edinburgh Sect. A 147 (2017) where only the case $s=0$ is considered. Moreover, thanks to monotonicity properties of the solutions, we separate two branches of non-radial solutions.