Non-existence results for fourth order Hardy–Hénon equations in dimensions 2 and 3

• Published in: Journal of Differential Equations 2024-07, Vol.397, p.55-79
• Main Authors: Tran, Thi Ngoan, Ngô, Quốc Anh, Tran, Van Tuan
• Format: Article
• Language: English
• Subjects: Classical solution; Dimensions 2 and 3; Distributional solution; Existence and non-existence; Hardy–Hénon equation; Pointwise estimate
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2024.02.057

We consider the fourth order Hardy–Hénon equationΔ2u=|x|σupin Rn with n≥2, σ>−4, and p>1. This is the fourth order analogy of the second order equation −Δu=|x|σup, known as the Hardy–Hénon equation, which 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 fourth order equation, the assumption n≥4 is often assumed. In this work, we are interested in classical solutions to the equation in the case of low dimensions, namely 2≤n≤3. Here by a classical solution u we mean u belongs to the class C(Rn)∩C4(Rn∖{0}) if σ