The Existence Of Solution Of Nonlocal Kirchhoff Equation

In recent years, with the development of nonlocal operator, we focus on the research about the existence of solution of one nonlocal kirchhoff equation. in this paper, we consider the following nonlocal kirchhoff equation:in which A?is a nonlocal operator.In the first chapter, firstly it mainly describes the research background and research status of the nonlocal differential equation, introducing the main contents and some conclusions of this paper.In the second chapter, firstly it mainly introduces the fractional Sobolev space?fractional Sobolev embedding theorem, secondly it introduces the nonlocal operator?the kernel?nonlocal valume-contrained problems?nonlocal energy space by introducing the nonlocal operator, pointing out when considering the special kernel, we can prove the nonlocal energy space is equivalent to the fractional Sobolev space, so we can define the fractional Sobolev space H?2(?) as our working space.In the third chapter, firstly by using variational methods, we constructing the corresponding energy functional, converting the research problem into solving the corresponding energy functional's critical point problem. secondly we separately discuss the following four situation about p, ?:(1) 3 < p <n+?n-?, ? > 0(2) p = 3, 0 < ? < c4(3) 3 ? p <n+?n-?,? < 0(4) p = 3, ? ? c4.( in which c is the best Sobolev embedding constant). for the first three situation, similarly we can get the energy functional's critical point by firstly verifying complacently Mountain pass theorem and secondly satisfying PS condition, so the equation has the nontrival solution. the creation of the paper is the research about the situation(3). for the last situation, we can immediately use proof by contradiction finding the equation only has trival solution.In the fourth chapter, it illustrates and researches some prospects of the topic, which including well-posedness of solution, the boundary conditions, the definition domain of equation and the nonlocal hyperbolic kirchhoff equation.