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Liouville theorems for Kirchhoff equations in RN

Authors: 

Nhat Vy Huynh, Phuong Le* and Dinh Phu Nguyen

Source title: 
Journal of Mathematical Physics, 60(6): 061506, 2019 (ISI)
Academic year of acceptance: 
2019-2020
Abstract: 

This paper is devoted to the nonexistence of nontrivial weak solutions for the Kirchhoff equation 17.png in 18.png. We prove that the equation has no weak solution if a ≥ 0, b > 0, q ≤ −2, and f is a positive, convex, nondecreasing function. If only b ≠ 0 and f is a non-negative function, we establish the nonexistence of weak solutions u satisfying 19.png. This implies that the equation has no weak solution when N ≤ 2 and f is a positive function. We also show that the equation has no stable weak solution in dimension 20.png if f(u) = eu, a ≥ 0, and b > 0.