Existence And Multiplicity Of Solutions For Fractional Differential Equations With Critical Nonlinearity

In this thesis,we mainly study some kinds of fractional differential equations with critical exponent by means of variational methods,we obtain the existence of nontrivial solution and multiple solutions respectively under suitable conditions.The thesis consists of four chapters.In the first chapter,we give the background and significance of nonlocal partial differential equation,introduce the structure of the dissertation and some preliminaries.In chapter 2,we study two kinds of fractional Kirchhoff type problems with critical nonlinearity.First,we study a class of Kirchhoff type problems involving fractional p&q problem with critical Sobolev-Hardy exponents and sign-changing weight functions:where b:??R is a sign-changing function.By using the fractional version of concentration compactness principle together with mountain pass theorem,we obtained the multiplicity of solutions for this problem.Second,we study the existence of weak solutions for the fractional p-Laplacian equation with critical nonlinearity in RN where potential function V(x)has critical frequency,that is minx?RN V(x)= 0.By us-ing fractional version of concentration compactness principle together with variational method,we obtained the existence and multiplicity of solutions for the above problem.In chapter 3,we study two kinds of fractional Kirchhoft type problems with critical nonlinearity and electromagnetic fields.Some difficulties arise when dealing with this problem,because of the appearance of the magnetic field and of the nonlocal nature of the fractional Laplacian.Therefore,we need to develop new techniques to overcome difficulties induced by these new features.First,we study the fractional p&q-Laplacian problem with electromagnetic fields and critical nonlinearity:where(-?)rs,A is the r-fractional magnetic operator.We prove the existence of in-finitely many solutions of this problem which tend to zero by using a new version of the symmetric mountain-pass lemma due to Kajikiya.Second,we study a class of the p-fractional Schrodinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity:By using the variational methods,we obtain the existence of mountain pass solutions u? which tend to the trivial solutions as ??0.Moreover,we get m*pairs of solutions for the problem in absence of magnetic effects under symmetric assumptions.In chapter 4,we study degenerate fractional Kirchhoff equations with critical non-linearities,this is,the Kirchhoff function M(0)= 0 in the degenerate case.The research on degenerate fractional Kirchhofi equations is also of great practical significance,but it presents a lot of difficulties in proving the(PS)c condition.To overcome this difficulty,we fix parameter ? under a suitable threshold strongly depending on assumptions for Kirchhoff function M.Because of the energy functional neither upper nor lower bound.we will use the truncation method to solve this difficulty.We prove the existence of infinitely many solutions of this problem which tend to zero by using a new version of the symmetric mountain-pass lemma due to Kajikiya.