Study On The Existence Of Solutions Of Some Fourth - Order Elliptic Problems

In this paper, we study the existence of solutions for three fourth-order elliptic problems.In Chapter 1, we study the fourth-order nonlinear problems in R^.where N≥1, △2:=△ (△) is the biharmonic operator, fu(x,u,v) and fv(x,u,v) are the gradients of F(x,u,v). V1(x), V2(x) and F(x,u,v) are positive functions. Under some assumptions on fu(x, u, v), fv(x, u, v), we prove the existence of many nontrivial high and small energy solutions by variant Fountain theorems.In Chapter 2, we study the following fourth-order nonlinear elliptic problem: where Ω(?) RN (N>4) is a smooth bounded domain, △ is the laplace operator and △2 is the biharmonic operator,2*=2N/N-4, c is a constant and f(x, u) is a continuous function on Ω × R. And the fourth-order quasilinear elliptic problem on Ω: where g1 and g2 are continuous functions in R. Based on the Limit Index Theory, The existence of multiple solutions for the fourth-order nonlinear problem are obtained and We generalize the result to the fouth-order quasilinear problem.