Ground State Solution for a Fourth Order Elliptic Equation of Kirchhoff Type with Critical Growth in ℝ N

In this paper, we consider the following fourth order elliptic Kirchhoff-type equation involving the critical growth of the form Δ2 u – (a + b Ð ℝN j∇uj 2 dx)Δu + V(x)u = {Iα ∗ F(u)}f(u)+λjuj 2∗∗−2 u, in ℝN, u ∈ H2(ℝN), ( where a > 0, b ≥ 0, λ is a positive parameter, α ∈ (N − 2, N), 5 ≤ N ≤ 8, V : ℝN ⟶ℝ is a potential function, and Iα is a Riesz potential of order α. Here, 2∗∗ = 2N/(n – 4) with N ≥ 5 is the Sobolev critical exponent, and Δ2 u = Δ(Δu) is the biharmonic operator. Under certain assumptions on V(x) and f(u), we prove that the equation has ground state solutions by variational methods.