Existence And Multiplicity Of Solutions For The Fractional Laplacian Problems

The dissertation mainly focuses on the existence and multiplicity of solutions for the fractional Laplacian problems by using the variational techniques.First of all,we study the nonlocal critical problem.Suppose nonlinear term satisfies certain growth conditions,for the critical problem,we get the existence of single solution using both the Mountain Pass theorem and Linking theorem.The results generalize the main results of Servadei et al [Trans.Amer.Math.Soc.,2015],[Rev.Mat.Complut.,2015],and include Theorem1.1 of [Adv.Nonlinear Anal.,2013].Above all,we prove the existence of multiple solutions for the critical problem.This result generalizes the main results about Sang [Nonlinear Anal.,1994],Fiscella et al [Bull.Sci.Math.,2016].Then,we also study the nonlocal subcritical problem.The existence of infinitely many solutions are obtained of the subcritical problem by the Symmetric Mountain-Pass theorem when nonlinear term satisfies the appropriate conditions.This result generalizes the main results about Kajikiya et al [J.Math.Anal.Appl.,1990],Zhang et al[Nonlinearity.,2015].