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In this paper, a critical fourth-order Kirchhoff type elliptic equation witha subcritical perturbation is studied. The main feature of this problem is thatit involves both a nonlocal coefficient and a critical term, which bringessential difficulty for the proof of the existence of weak solutions. When thedimension of the space is smaller than or equals to $7$, the existence of weaksolution is obtained by combining the Mountain Pass Lemma with some delicateestimate on the Talenti's functions. When the dimension of the space is largerthan or equals to $8$, the above argument no longer works. By introducing anappropriate truncation on the nonlocal coefficient, it is shown that theproblem admits a nontrivial solution under appropriate conditions on theparameter.

Existence of nontrivial solutions to a fourth-order Kirchhoff type elliptic equation with critical exponent