Liouville theorems for Kirchhoff equations in R^N

This paper is devoted to the nonexistence of nontrivial weak solutions for the Kirchhoff equation  in . 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 . 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  if f(u) = e^u, a ≥ 0, and b > 0.