Existence Of Solutions For Two Kinds Of Harmonic Equations With Critical Exponents

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:Δ2u-μV(x)u=|u|2*-2 u+θh(x),u∈H02(Ω).(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 Mη(usually called Nehari manifold).Secondly,the set Mη is divided into three parts Mη+,Mη0 and Mη-by using fibering maps.Moreover,we prove the existence of minimum in Mη+ and Mη-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.