Existence Of Nontrivial Solution For The Biharmonic Equation

It is a core of the theory of research field of differential equation to study multiplic-ity results for differential equations under boundary conditon. Also,it is one of the keysubjects of the research contents of this field.This text chooses the biharmonic equation, a sort of elliptic equations with high or-ders,utilizes the methods such as variational reduction method and critical point principlemainly to study solvability and multiplicity of solutions of the biharmonic equations andshows the new development of the PDEs.1. The first chapter introduces the development and the background of the new meth-ods about the PDEs, we will know the results in this area. We also get the main resultsof this paper.2. In the second chapter we introduces some important definitions and throrems.3. In the third chapter we investigate a sort of biharmonic equation with criticalexponentΔ²u+cΔu=f(x)u^q-1-u^p-1. we get the two positive solutions of the equationusing the Hardy inequality and the Mountain Pass Lemma.4. In the fourth chapter we study the fourth-order semilinear elliptic equationΔ²u+a²Î”u=bg(x, u). We use the Three Solution Theorem and know the equation has at leastthree nontrival solutions.