The Existence And Multiplicity Of Solution For Two Classes Of P-Laplacian Biharmonic Equation

Biharmonic equation and p-Laplacian equation are the important research content of modern partial differential equations.The biharmonic equations were widely used in optics,plasma physics,elastic mechanics,engineering and other fields.The p-Laplacian equation-s were born in many physical phenomena,such as newtonian fluids,nonlinear elasticity,glaciology and oil exploration.In recent years,the biharmonic equations with p-Laplacian have attracted more and more attention from scholars.In this thesis,under appropriate assumptions to the nonlinear terms,the existence and multiplicity of solution for two classes of p-Laplacian biharmonic equation are studied by using mountain path theorem,fountain theorem and Nehari manifold methods.The thesis consists of two chapters.In chapter 1,we discuss the following p-Laplacian biharmonic equation with sign-changing potential Wherep ?[2,2 N(N-4)),N?5,?2u=?(?u),?pu=div(|?u|p-2?u),f? C(RN×R,R),V?C(RN)is sign chaning.When the nonlinearity f satisfies the appropriate assumptions,the existence of nontriv-ial solutions and ground state solutions to the above equation is considered.Specifically,a Banach space,the work space of this paper,is firstly constructed by using the properties of the sign-changing potential.Secondly,the mountain path type critical point is obtained for the energy functional corresponding to the equation.Then by comparing the least energy of the Nehari manifold with the energy of mountain path type critical point,the existence of the ground state solutions to the equation is showed.In chapter 2,Specifically,by using mountain path theorem,the existence of nontrivial solutions to the following p-Laplacian biharmonic equations with quasilinear term are mainly studied,then by using the fountain theorem,the existence of infinitely many high energy solutions to the equation is obtained,where p?[4,6],?pu=div(|?u|p-2?u),?2u=?(?u),f ? C(R3 × R,R),V ? C(R3)is sign-chaning and bounded below.