Bifurcation analysis of the Hardy-Sobolev equation

TL;DR

This paper proves the existence of multiple non-radial solutions to the Hardy-Sobolev equation, extending previous results for the case s=0 and analyzing bifurcation and solution branches.

Contribution

It extends bifurcation analysis of the Hardy-Sobolev equation to cases with s>0, identifying multiple non-radial solutions and their solution branches.

Findings

Existence of multiple non-radial solutions for the Hardy-Sobolev equation.

Extension of previous results from s=0 to s in [0,2).

Separation of solution branches based on monotonicity properties.

Abstract

In this paper, we prove existence of multiple non-radial solutions to the Hardy-Sobolev equation $$\begin{cases} -\Delta u-\displaystyle\frac \gamma{|x|^2}u=\displaystyle\frac{1}{|x|^s}|u|^{p_s-2}u & \text{ in } \mathbb{R}^N\setminus\{0\},\\ u\geq 0, & \end{cases}$$ where $N\geq 3$, $s\in[0,2)$, $p_s=\frac{2(N-s)}{N-2}$ 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.