Existence Of Nontrivial And Multiple Solutions For A Navier Boudary Value Problem Involving The P-Biharmonic Equation

In this paper, we consider the existence and multiplicity solutions for a Navier boundary value problem involving the p-biharmonic equa-tion in W2.p(Ω)∩W01.pp(Ω). This paper contains three parts as following:In chapter one, we will introduce the background of p-biharmonic equation, main contents and preliminaries.In chapter two. we consider the following nonlinear Navier bound-ary value problem involving the p-biharmonic operator where λ is a positive parameter, p>max{1,N/2}, Ω is an open bound-ed subset of Rn with a smooth enough boundary (?)Ω,Δ is the usual Laplacian operator,f:R→R is a continuous function.We consider the nonlinearity satisfies certain growth condition only in some neighborhood of s=0, by using critical point theory, we obtain two nontrivial solutions of the equation (P1), one is positive solution and the other is negative solution.In chapter three, we consider the following nonlinear Navier bound-ary value problem involving the p-biharmonic operator the existence of a nontrivial solution, where λ is a positive parameter,1<p<∞, Ω is an open bounded subset of RN with a smooth enough boundary (?)Ω3, and f is a suitable continuous function defined on Ω×R.In the problem of (P2), we can obtain at least one nontrivial solution when the nonlinearity f is super p-linear, subcritical and quasimono-tone, without assuming the famous Ambrosetti-Rabinowitz condition.