The research of existence,multiplicity and other related properties of solutions,for elliptic problems,has an important significance both in theory and reality.We study two different elliptic differential equations,namely fourth-order semilinear equations with critical exponents and descriptive singularities and the quasilinear p-biharmonic equations with critical Sobolev-Hardy terms by using the variational method in this paper.This paper focuses on the following fourth-order semilinear equations with critical exponents and descriptive singularities:?2u-?V(x)u=|u|2*-2 u+?h(x),u?H02(?).(1)By constructing the appropriate minimization sequence and using the variational method,the sufficient conditions for the existence of multiple positive solutions for the problem(1)are obtained.And we study the following quasilinear p-biharmonic equations with critical Sobolev-Hardy terms:(?)The existence theorem of the solutions to the above problem is established by means of the Ekeland variational principle.Firstly,to guarantee the variational functional is bounded from below,we restrict it on a set M?(usually called Nehari manifold).Secondly,the set M? is divided into three parts M?+,M?0 and M?-by using fibering maps.Moreover,we prove the existence of minimum in M?+ and M?-by studying the properties of above two subsets.Finally,by using implicit function theorem,we get that there exists a minimizing sequence{un},such that the(PS).conditions hold,when the parameters satisfy certain conditions.Therefore,we show the existence of the solutions for the problem(2).The conclusions will provide a theoretical basis for judging the structure and properties of the solutions. |