Existence And Multiplicity Of Solutions For Some Nonlinear Differential

In this paper,we mainly study existence and multiplicity of solutions for some nonlinear differential equations including the wave equation and elliptic equation.The main mathematical tool we used is the topological degree theory in nonlinear analysis.Firstly,we consider the Dirichlet-Neumann boundary value problem of nonlinear variable coefficient wave equation.By analyzing spectral characters of variable coefficient wave operator,we prove that the variable coefficient wave operator is invertible and its inverse is compact.Thus,by using the Leray-Schauder degree theory,we can obtain the existence and multiplicity of time periodic solutions.Secondly,we consider the Neumann boundary value problem of constant coefficient wave equation.By introducing a suitable subspace,we prove that wave operator is invertible and its inverse is compact in the subspace.Then,by using the topological degree theory,we obtain the existence of multiple time periodic solutions.Thirdly,we consider the Dirichlet-Neumann boundary value problem of a variable coefficient elliptic equation.By analyzing the variable coefficient elliptic operator,under the suitable assumption on the coefficients,we obtain the invertibility of variable coefficient elliptic operator and the compactness of its inverse.Thus,we also obtain the existence of multiple solutions by using the topological degree theory.