Classification results for Kirchhoff equations in RN. Issue 7 (3rd July 2019) - Le, Phuong; Huynh, Nhat Vy; Ho, Vu
ABSTRACT: We study the Kirchhoff equation of Hardy-Hénon type− a + b ∫ R N | ∇ u | 2 d x Δ u = | x | p | u | q − 1 u in   R N, where constantsa, b ≥ 0 and a + b >0. We show that this equation has no positive solution ifp ≤ − 2 or1 < q < ( N + 2 p + 2 ) / ( N − 2 ), whereN ≥ 3 . In the critical caseq = ( N + 2 p + 2 ) / ( N − 2 ) where− 2 < p ≤ 0, 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 dimensionN < ( 2 ( p + 2 ) ( q + q ( q − 3 ) ) ) / ( 3 ( q − 1 ) ) + 2, where q >3.