Existence Of Solutions For Boundary Value Problems Of Fourth Order Kirchhoff Type Equations

In this paper,we mainly study the existence of solutions to boundary value problems of fourth order Kirchhoff type equations.The paper consists of three parts.The first chapter briefly describes the historical background,research status,preparatory knowledge and main conclusions.In the second chapter,we prove the existence of nontrivial solutions for the Dirichlet boundary value problems of the following fourth-order Kirchhoff type e-quations by the variational method where Ω(?)R6 is a bounded open region with a smooth boundary(?)Ω,a,b,λ are positive parameters.The key of this chapter is to prove that the energy functional of equation(1)satisfies the PS condition,and then prove the existence of the nontrivial critical point of the energy functional of equation(1)by the Mountain Pass Lemma.In the third chapter7 we prove the existence of positive solutions for the Dirichlet boundary value problems of the following fourth-order Kirchhoff type equations by using the continuity method with a,b are positive parameters,B is a ball in RN(N≥4),p ∈(0,4*)\{1} and 0<α<4*-1/2.Where 4*=N+4/N-4 for N≥5 and 4*=+∞ for N=4.The emphasis of this chapter is to obtain a priori estimate of the solution.Therefore,First,we obtain the maximum modulus estimates by means of the Liou-ville Theorem and the blow-up method,and then C4,α estimates by means of the regularity theory.Before that,we will also prove the influence of nonlocal terms on the positive solution set in the case of h(x,u,▽u)=0.