Study On Solutions For Two Classes Of Kirchhoff Fractional Equations On The Heisenberg

Kirchhoff-type equation is a kind of non-local partial differential equation used to describe the length transformation of the transverse vibration of a telescopic rope.It has been widely used in the practical problems of cosmology,non-Newton mechanics,plasma problems and so on.Heisenberg group is a special non-commutative Lie group,which plays an important role in many fields such as quantum mechanics,partial differential equation theory,number theory and so on.In this thesis,inspired by the study of Sobolev spaces and Kirchhoff-type fractional Laplace equations in Euclidean spaces,we will examine the existence and multiplicity of solutions for two classes of Kirchhoff-type fractional p-Laplace problem on the Heisenberg group.Firstly,the thesis introduces the related basic theories on the Heisenberg group,including the operation rules of the group,the expansion transformation,the Kohn-spencer Laplace operator and so on.Next,we introduce a new class of fractional Sobolev-type spaces on the Heisenberg group,and prove that the space has uniform convexity,reflexivity and their embedding relations.Secondly,for a class of Kirchhoff fractional p-Laplace equation on the Heisenberg group,the existence of solutions for the sublinear and suplinear cases is studied by weakening the structural assumption of the right-hand term.Palais-Smale conditions are proved by the concentration-compactness principle.Then,using the variational method and mountain pass theorem,the thesis proves the existence of nontrivial and nonnegative minimum solution and mountain pass solution of the equation.Finally,we study the multiplicity of solutions for a class of Kirchhoff-type fractional p-Laplace equations with weighted functionA（（?））u^r-1u,and prove the existence of infinite many solutions by means of the fountain theorem and the dual fountain theorem.