Classification results for Kirchhoff equations in RN. Issue 7 (3rd July 2019)

• Record Type: Journal Article
• Authors: Le, Phuong; Huynh, Nhat Vy; Ho, Vu
• Is Part Of: Complex variables and elliptic equations. Volume 64:Issue 7(2019)
• Page Start: 1146
• Page End: 1157
• Publication Date: 2019-07-03
• Subjects: P. Pucci; Kirchhoff equation -- nonexistence -- classification -- positive solution -- stable solution -- critical exponent; 35A02 -- 35B53 -- 35B09 -- 35B33 -- 35B35; Functions of complex variables -- Periodicals; Differential equations, Elliptic -- Periodicals; 515.905
• DOI: 10.1080/17476933.2018.1505874
• Languages: English
• ISSNs: 1747-6933

Abstract

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.