Existence, Multiplicity And Concentration Of Solutions For Several Classes Of Fractional Elliptic Equations With Critical Exponent

The fractional differential equations have profound physical background and rich theoretical connotation.Fractional differential equations are closely related to geom-etry,functional analysis,quantum mechanics,probability theory,and are ones of the most active research fields in the last decade.In the study of fractional differential equations,the nonlocal problems are more difficult than the local problems.In the pi-oneering article of Caffarelli and Silvestre,the basic properties of non-local equations,such as regularity and extremum principle,have been established,which lay the foun-dation for the use of various nonlinear analysis tools.In this thesis,we study the existence,multiplicity and concentration of solutions for several fractional elliptic equations involving critical exponent.The main contents are as follows.In the first part,without the(AR)condition and monotonicity condition on the nonlinearity,we study the existence of ground states of fractional Schr?dinger equation with critical exponent.Because of the loss of compactness and difficulty to get the boundedness of(PS)-sequence,we apply the monotonicity trick and auxiliary equations to obtain a special bounded(PS)-sequence.Then by using of the decomposition of bounded(PS)-sequence,we recover the compactness and then obtain the existence of ground states.In the second part,we study the existence and concentration of the solutions for the singularly perturbed fractional equation with critical exponent.Because of the loss of compactness and without the(AR)condition and monotonicity condition,by the Moser iteration argument in the whole space R~N,we get a priori L~?-estimate of ground states to the limit problem.Then we reduce the original problem to a subcritical problem by the truncation technique.By use of the existence and concentration of the subcrit-ical problem,we obtain the existence and the concentration of the critical singularly perturbed problem.In the third part,we study the existence and multiplicity of solutions for fractional Kirchhoff equations with critical exponent.Because of the presence of the Kirchhoff term,in high dimension N>4s,mountain pass geometry and the(AR)condition do not hold.We use the perturbation approach to get a bounded(PS)-sequence.Then we obtain the existence of the solutions and the asymptotic behavior of the solutions according to the parameters.In addition,by virtue of the truncation technique,the concentration compactness principle and a linking theorem,we obtain the multiplicity of solutions for a class of fractional Kirchhoff equations.In the final part,we discuss the existence of ground states of the fractional Choquard equation with Hardy-Littlewood-Sobolev critical exponent.In order to get the upper estimate of critical problem,we get the achieved function for the correspond-ing Sobolev imbedding and the corresponding best imbedding constant value.Using the idea of approximation,we get the bounded(PS)-sequence for the critical problem.Then,we present a splitting lemma.Together with the compactness lemma,we obtain the existence of nonnegative ground states for the critical problem.