Multiplicity Solutions Of A Nonlinear PDE With Nonlinearity Crossing Eigenvalues

Through the variational method we investigate the multiple solutions of a nonlinear PDE: We proved that when the nonlinear part b crosses three eigenvalues, the above equation has at least six nontrivial solutions.In Chapter1, we introduce the beginning and development of variational method and the way to solve some variational problem.In Chapter2, we list some required definitions in the Banach space and Sobolev space, and important lemmas in the critical theory.In Chapter3, we change the question of finding the solutions of the nonlinear PDE to the variational problem and make detailed analysis of it.In Chapter4, first, we prove that the functional satisfies Three holes Torus-Sphere variational linking inequality, using the conclusion of [1] we can find four mountain pass type critical point. Second, we can find the fifth and sixth mountain pass type critical points through the finite dimensional reduction method. Above all we have six nontrivial critical points which means the PDE has at least six solutions.