Classification results for Kirchhoff equations in R^N

We study the Kirchhoff equation of Hardy-Hénon typewhere constants  and a+b>0. We show that this equation has no positive solution if  or , where . In the critical case  where , we provide a necessary and sufficient conditions on a and b for the equation to have positive solutions and we classify all such solutions. We also prove the nonexistence of nontrivial stable solutions of the equation in dimension , where q>3.