Solutions To Fourth-Order Elliptic Equation With Nonlocal Term

This thesis focus on Kirchhoff type elliptic equations.Firstly,we study the following Kirchhoff type elliptic equation where Δ2=Δ(Δ)is the bi-harmonic operator,a,b>0 are constants,and V∈C(R3,R).Under the suitable conditions,two solutions,namely,a mountain-pass solution as well as a ground state solution,are obtained by taking advantage of the variational methods.Secondly,we mainly study the Kirchhoff type elliptic equation with perturbation term where a,b>0,λ>0 are constants,V ∈ C(R3,R).Under the suitable assumptions,forλ>0 small enough,combining the Ekeland variational principle and the Mountain-pass theorem,we prove that the equation possesses at least two nontrivial solutions.Then we will demonstrate that the equation has infinitely many high-energy solutions via Fountain theorem when g=0,which improves and generalizes some existing results.