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

Abstract

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<0 and C4(Rn) if σ≥0. We show that if 2≤n≤3 then the equation admits no classical solution. In fact, we are able to provide a single treatment for 2≤n≤4.