Existence Of Solutions For A Class Of Biharmonic Equations With Singular Potential

The study of the boundary value problem for biharmonic equation is one of the hotspots in the study of the boundary value problem for elliptic equation.In the case of distinct non-linear terms,using variational method to study the existence,multi-solution and eigenvalue problem of the high energy solution,non-trivial solution and sign-changing solution of biharmonic equation has become one of the frontier topics in the study of non-linear partial differential equation.The existence of solutions for the following fourth-order nonlinear elliptic equations(i.e.biharmonic equations)is discussed:(?)where N?5,?~2 is the biharmonic operator,V(x)is a singular potential,andf?C(R~N×R,R).The specific research contents are as follows:In Chapter 2,we mainly consider that the problem has at least one non-trivial solution in the whole space R~N(N?5)and under the assumption that the nonlinearity term satisfies weaker conditions by combining Morse theory with local linking method in critical point theory.Then we prove that the energy functional satisfies the symmetric mountain path lemma by constructing a strongly deformed shrinking core with implicit function and combining Hardy-Rellich inequality,which proves the multiplicity of nontrivial solutions of the problem.In Chapter 3,this part is based on Chapter 2 to further explore and analyze the solutions of the problem.The symbols of nontrivial solutions are discussed by using the properties of the invariant set of descending flow.The existence of positive solutions,negative solutions and sign-changing solutions of the problem are obtained.The acquisition of these results is an effective extension and supplement to the results in some literatures.