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

Abstract

In this paper, we consider the following fourth order elliptic Kirchhoff-type equation involving the critical growth of the form Δ 2 u − a + b ∫ ℝ N ∇ u 2 d x Δ u + V x u = I α ∗ F u f u + λ u 2 ∗ ∗ − 2 u , in ℝ N , u ∈ H 2 ℝ N , where a > 0 , b ≥ 0 , λ is a positive parameter, α ∈ N − 2 , N , 5 ≤ N ≤ 8 , V : ℝ N ⟶ ℝ is a potential function, and I α is a Riesz potential of order α . Here, 2 ∗ ∗ = 2 N / n − 4 with N ≥ 5 is the Sobolev critical exponent, and Δ 2 u = Δ Δ u 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.