The Existence Of Solutions For Two Classes Of Nonlinear Partial Differential Equations

In this paper,the existence of solutions to two types of nonlinear partial differential equations is studied.On the one hand,we divided variational problems into three categories by spectra of operator A,and for a class of complex cases of σe(A)≠?,we used saddle point reduction and the index pair(iA0(B),νA0(B))to obtain some critical point theorems for abstract operator equations without compactness assumption,and proved the existence of the following periodic solutions of one-dimensional wave equations.where T>0,S1:=R/TZ,f:[0,π]×S1×R→R.On the other hand,we considered the following semi-linear Schrodinger equation using the relative Morse index.We developed the concept of Maslov type index(μ(M),υ(M))for(4-1)by the relative Fredholm index theory,and displayed the relationship with other index Theories,obtained the existence of nontrivial solutions of(4-1).