The Existence Of Dirichlet Problem Solutions In Nonlocal Kirchhoff Differential Equations

Recently, nonlocal partial differential equation has a very wide application in many areas, such as continuous media mechanics, the phase image, population dynamics and the game theory, etc.In the field of nonlocal equation in theoretical physics, financial system and fluid mechanics has been widely used, the fractional order differential equations than integer order partial differential equations can more accurate to depict the natural phenomenon, to the simulation of the dynamic process of life better. Fractional order differential equation is the promotion of the integer order equations.This paper discusses the existence of the solutions to nonlocal Kirchhoff type equations by using variational methods.We consider the problem as follows:where ? is a bounded domain in Rn,n? 4,u ? H?2(D),0 < ? < 2.The nonlocal operator A?is defined as follows:In the first chapter, we mainly introduce the background of the nonlocal partial differential equation, significance and research status in China and abroad, also the main contents and conclusions in this paper.In the second chapter, we introduce the basic knowledge of the functional analysis, and some preliminary knowledge of Sobolev spaces, nonlocal operator theory and variational method.In the third chapter, this article will prove functional when lambda satisfies the conditions of the Mountain Pass Lemma, to prove bounded sequence and meet the P-S and prove energy functional with critical point column and equation has a non trivial weak solution. Followed by lambda and full hours. By the solution of the minimum of the functional,we prove that the equation has at least two nontrivial weak solutions. To prove lambda when functional of the meet the Mountain Pass Lemma. It is proved that the equation has a nontrivial weak solution.In the fourth chapter, we discuss the expectation of this paper.Its content mainly includes: the definition of the local operator in unbounded region and the existence of the solutions and in the whole space, functional existence.