The Research Of The Problem About Solution And Eigenvalue Of Biharmonic Equations

This paper mainly dicusses the existence of nontrival solution, multiple solutions problem and eigenvalue problem for biharmonic equations with different forms. As we know, many results have been discovered which are compatible with forms, so we pay our attention to the generality and complexity of biharmonic equations. The main method is variational method and its flexible application in different equations.This paper mainly has three parts:the existence of nontrival solution for biharmonic equations, the multiple solutions for biharmonic equations with disturbance term and the eigenvalue problem for biharmonic equations. In the first part, the equation has general form which is gradual convergence. Because of the form of equation is not specific, so the process of solving the problem with general method is difficult. In this condition, we use the lower semi-continuity result. In the second part, the main problem is the multiple solutions for biharmonic equations with disturbance term. For the influence of disturbance term, the solution of equation may be not the same as the result of equations with no disturbance term. Jumping over obstacles about disturbance term is our focus. In the third part, the complexity of problem about eigenvalue springs from the complexity of equation. because the form of equation is general, so some method for special equation is not applying. Under this condition, the transfiguration of equation and applying Mountain pass lemma correctly are the key of solving this problem.