For the different forms of fourth order elliptic equations,and applies the Lax Milgram theorem and variational method are respectively applied to study two types of fourth order elliptic equations.In the first part of this article,Lax Milgram is used to verify the existence and uniqueness of a solution u on a Hilbert space H2,so that the bounded mandatory bilinear form is equal to any bounded linear functional on H2.Furthermore,it is proven that there exists a unique weak solution for a class of fourth order elliptic equations with a first order term.In the second part of this article,the variational method is used to solve another class of fourth order elliptic equations with second order terms of degree p.In the method,firstly,the weak solution of the equation is defined,then the corresponding functional is found out,and then the problem is transformed into the extremal element of the corresponding functional,the existence of the extremal element of the functional is proved,and finally the uniqueness of the weak solution is proved.
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