Exhaustive existence and non-existence results for Hardy–Hénon equations in $${{\,\mathrm{{\textbf{R}}}\,}}^n$$ R n

Yoshikazu Giga () and Quốc Anh Ngô ()
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Yoshikazu Giga: The University of Tokyo
Quốc Anh Ngô: Vietnam National University

Partial Differential Equations and Applications, 2022, vol. 3, issue 6, 1-38

Abstract: Abstract This paper concerns solutions to the Hardy–Hénon equation $$\begin{aligned} -\Delta u = |x|^\sigma u^p \end{aligned}$$ - Δ u = | x | σ u p in $${{\,\mathrm{{\textbf{R}}}\,}}^n$$ R n with $$n \ge 1$$ n ≥ 1 and arbitrary $$p, \sigma \in {{\,\mathrm{{\textbf{R}}}\,}}.$$ p , σ ∈ R . 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 $$p>1,$$ p > 1 , $$\sigma \ge -2,$$ σ ≥ - 2 , and $$n \ge 3$$ n ≥ 3 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.

Keywords: Hardy–Hénon equation; Lane–Emden equation; Existence and non-existence; Liouville theorem; Primary 35B53; 35J91; 35B33; Secondary 35B08; 35B51; 35A01 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s42985-022-00190-3

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