Existence Of Solutions For A Class Of Fractional Kirchhoff Type Equations With Variable Exponent

With the progress of science and technology,fractional partial differential equations have attracted more and more attention in the fields of mathematics,physics and seepage mechanics,while fractional Kirchhoff partial differential equations are a kind of meaningful nonlinear partial differential equations.Kirchhoff type equation is a partial differential equation with nonlocal properties abstracted from elasticity,Kirchhoff type term can be used to describe the length change of transverse vibration of retractable rope.It has a wide range of applications in aerospace,astrophysics,non-Newtonian mechanics and so on.Inspired by relevant research results,this paper will further extend the research results of fractional Kirchhoff type partial differential equations with constant index,deeply study the existence of solutions for a class of Kirchhoff type fractional partial differential equations with variable exponential growth conditions,and give some research results with theoretical significance.The specific research contents are as follows,Firstly,this paper will summarize the research background and development status of variable exponential function space,Kirchhoff function,fractional Laplace equation and variable exponential fractional Laplace equation.Secondly,considering the need to study the existence of solutions of Kirchhoff type fractional partial differential equations,this paper will introduce the basic theories and properties of variable exponential function space and fractional function space in detail.Finally,the energy functional is constructed by using the variational structure of the equation,and the definition of weak solution of the equation is given Considering that the energy functional does not have lower boundedness in ftractional Sobolev space,in order to overcome this difficulty,this paper limits the energy functional to the Nehari manifold.By proving the mandatory and lower boundedness of the energy functional and combining the relevant properties of the Nehari manifold and the energy functional,it is obtained that this kind of equation has at least two different nonnegative solutions.