The Existence And Multiplicity Of Solutions To Two Types Of Nonlinear Partial Differential Equations

In this paper,we mainly consider two problems.Firstly,we concern with the following elliptic equation:Where the primitive function of f(x,u)is either superquadratic or asymptotically quadrat?ic as |u|→∞,or subquadratic as |u|→0 By using variational method,e.g.the local linking theorem,Fountain theorem,and the Generalized Mountain Pass Theorem,we establish the existence and multiplicity results for the periodic solution and subharmonic solution.Secondly,we consider the following modified quasilinear fourth-order elliptic equation:where△~2=△(△)is the biharmonic operator,a＞O,b≥O,and,λ≥1 is a parameter,V∈C(R~3,R),f(x,u)∈C(R~3×R,R).V(x)and f(x,u)u are both allowed to be sign-changing.Under the weaker assumption lim(?),a sequence of high energy weak solutions for the above problem are obtained.