Instability of solutions to Kirchhoff type problems in low dimension

We study the Kirchhoff type problem

where p ≥ 2, Ω is a C¹ domain of  , 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.