Super Polyharmonic Property and Asymptotic Behavior of Solutions to the Higher Order Hardy-Hénon Equation Near Isolated Singularities

In this paper, we are devoted to studying the positive solutions of the following higher order Hardy-Hénon equation (Formula presented.) with an isolated singularity at the origin, where n>2m, α>-2m and m≥1 is an integer. For 1<p<n+2mn-2m, we prove the singularity and decay estimates of solutions. For n+αn-2m<p<n+2mn-2m with -2m<α<2m, we show the super polyharmonic properties of solutions near the singularity, which are essential in studying polyharmonic equations. By utilizing these properties, we classify the isolated singularities and establish the precise asymptotic behavior of solutions for the fourth order case. Furthermore, we also classify the isolated singularities at infinity and show a uniqueness theorem for the fourth order Lane-Emden equation.