Existence and non-existence results for the higher order Hardy-Henon equations revisited

Authors：Quoc Anh Ngo; Ye, Dong^*

Source：Journal des Mathematiques Pures et Appliquees, 2022, 163: 265-298.

DOI：10.1016/j.matpur.2022.05.006

• Higher-order Hardy-H?non equation
• Distributional solution
• Existence and non-existence
• Weak and strong
• super-polyharmonic property

This paper is devoted to the study of non-negative, non-trivial (classical, punctured, or distributional) solutions to higher order Hardy-Henon equations (-delta)(m)u = |x|(sigma)u(p) in R-n with p > 1. We show that the condition n- 2m 2m + sigma/p - 1 > 0 is necessary for the existence of distributional solution. For n >= 2m and sigma > -2m, we prove that any distributional solution satisfies an integral equation and weak super polyharmonic properties. We establish also some sufficient conditions for punctured or classical solution to be a distributional solution. As application, we show that if n >= 2m, sigma > -2m, there is no non-negative, non-trivial classical solution to the equation if 1 < p < n+2m+2 sigma/n-2m = :pS (m, sigma); and classical positive radial solutions exist for n > 2m, sigma > -2m and p > pS(m, sigma). Our approach is very different from most previous works on this subject, which enables us to have more understanding of distributional solutions, to get sharp results, hence closes several open questions.

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