The Problem Of The Existence Of Solutions Of Nonlinear Kirchhoff-type Differential Equations For Research

Since the seventies of the last century, modern variational methods(also known as large-scale variational methods) have a significant development with the invention of the Mountain Pass Theorem, the Saddle Point Theorem, mathematicians have widely applied it to solve nonlinear elliptic equations and obtained many new results.The Kirchhoff type equations were first proposed by Kirchhoff1883as an existence of the classical D'Alembert's wave equations for free vibration of elastic strings. They have great applications in many fields, such as non-Newtonian mechanics, cosmology and astrophysics, plasma problems andelasticity theory, so our study of these problems has a profound practical significance.In this paper, we apply the variational methods to study the existence and multiplicity of solutions for the nonlinear Kirchhoff type differential equations.On the one hand, for the bounded domain Ω of the integral area of the differential equations, we study the following type equation we will apply Linking Theorem, Pseudo-index theory to abtain the existence of the solutions of the above differential equations.On the other hand, when the integral area of the eqations is an unbounded domain, like the following type equations we study it with Symmetrical Mountain Pass Theorem and the Fountain Theorem to present two results of existence of infinitely many large energy solutions of the Kirchhoff type differential equations, respectively.