Exhaustive existence and non-existence results for Hardy–Hénon equations in $${{\,\mathrm{{\textbf{R}}}\,}}^n$$
Yoshikazu Giga, Quốc Anh Ngô
Abstract
Abstract This paper concerns solutions to the Hardy–Hénon equation $$\begin{aligned} -\Delta u = |x|^\sigma u^p \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>x</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mi>σ</mml:mi> </mml:msup> <mml:msup> <mml:mi>u</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> in $${{\,\mathrm{{\textbf{R}}}\,}}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mrow> <mml:mspace/> <mml:mi>R</mml:mi> <mml:mspace/> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> with $$n \ge 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and arbitrary $$p, \sigma \in {{\,\mathrm{{\textbf{R}}}\,}}.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>σ</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mspace/> <mml:mi>R</mml:mi> <mml:mspace/> </mml:mrow> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> This equation was proposed by Hénon in 1973 as a model to study rotating stellar systems in astrophysics. Although there have been many works devoting to the study of the above equation, at least one of the following three assumptions $$p>1,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> $$\sigma \ge -2,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo>≥</mml:mo> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> and $$n \ge 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> is often assumed. The aim of this paper is to investigate the equation in other cases of these parameters, leading to a complete picture of the existence/non-existence results for non-trivial, non-negative solutions in the full generality of the parameters. In addition to the existence/non-existence results, the uniqueness of solutions is also discussed.