Ground State Solution for a Fourth Order Elliptic Equation of Kirchhoff Type with Critical Growth in ℝN - Li Zhou; Chuanxi Zhu (Advances in Mathematical Physics)
In this paper, we consider the following fourth order elliptic Kirchhoff-type equation involving the critical growth of the form Δ2u−a+b∫ℝN∇u2dxΔu+Vxu=Iα∗Fufu+λu2∗∗−2u,in ℝN,u∈H2ℝN, 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 Δ2u=ΔΔu is the biharmonic operator. Under certain assumptions on Vx and fu, we prove that the equation has ground state solutions by variational methods.