Instability of solutions to Kirchhoff type problems in low dimension

Abstract

We study the Kirchhoff type problem  −m ( Ω w1|∇u| dx ) div(w1|∇u|∇u) = w2f(u) in Ω, u = 0 on ∂Ω, where p ≥ 2, Ω is a C domain of R , w1, w2 are nonnegative functions, m is a positive function and f is an increasing one. Under some assumptions on Ω, w1, w2, m and f , we prove that the problem has no nontrivial stable solution in dimension N < N. Moreover, additional assumptions on Ω, m or the boundedness of solutions can boost this critical dimension N to infinity.