Existence Of Solutions To Several P-Kirchhoff Type Differential Equations

Many problems in the natural science and engineering ares can be described by partial differential equations.p-Kirchhoff,as the most fundamental partial dif-ferential equations,because of its strong practical application background,has been received extensive attention,of large number of doemstic and foreign researchers.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 Banach fixed theorem,the fixed point index the-orem,the Linking theorem to study the existence of solutions for the nonlinear p-Kirchhoff type differential equations.The dissertation contains four 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 in demension three:where ?(?)R3 is a bounded domain with smooth boundary(?)?,a>0,b>0 and?>0 is a parameter.By using the Banach fixed theorem,under proper conditions,we obtain the problem has a positive solution.In chapter 3,we consider the following p-Kirchhoff type problem:Where ?(?)RN(N = 1,2,3)is a bounded domain with smooth boundary(?)?,a>0,b ? 0.By using the fixed index theorem,under proper conditions,we obtain the problem has a positive solution.In chapter 4,we consider the following p-Kirchhoff type problem:Where ?(?)RN(N = 1,2,3)is a bounded domain with smooth boundary(?)?,,a>0,b>0.By using the Linking theorem,under proper conditions,we obtain the problem has a sign-changing solution.