Multiple Solutions Of Kirchhoff Type Differential Equations With Singular Nonlinearity

Since the seventies of the last century, modern variational methods (also known as large-scale variational methods) has a significant development with the invention of the Mountain Pass Theorem, the Fountain Theorem, Mathematicians have wide-ly applied it to solve nonlinear elliptic equations and obtained many new results.The Kirchhoff type equations were first proposed by Kirchhoff 1883 as an ex-istence of the classical D'Alembert's wave equations for free vibration of elastic strings. They have great applications in many fields, such as non-Newtonnian me-chanics, 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, the concentration compactness principle, the Nehari method to study the existence and multiplicity of solutions for the nonlinear Kirchhoff type differential equations. The dissertation contains three chapters.In chapter 1, we mainly introduce the research status of Kirchhoff type dif-ferential equation and some basic knowledge and symbols commonly used in this paper.In chapter 2, we consider the following Kirchhoff type problems with singular-ity and critical growth in demension four: where ? (?) R4 is a bounded domain with smooth boundary (?) ?,?,?,?>0,b?0 and ? ? (0,1). By using the variational and perturbations methods, we obtain the problem has at least a positive solution. Furthermore, by the concentration compactness principle, we obtain the problem has at least two different positive solutions.In chapter 3, we consider the following Kirchhoff type problem with singularity and nonlinearity: where ? (?) R3, is a bounded domain with smooth boundary (?)?,0 ? ?,? ? (0,1), f(x),h(x) ? L~?(?) and f(x),h(x) are measurable functions, a>0,b?0,?? 0,?>0. By using the variational methods, we can obtain the uniqueness and existence of positive solutions of the problem. Furthermore, by the Nehari method, we can obtain the multiplicity of positive solutions of the problem.