Eigenvalue Problem Of Fractional Laplacian Operator With Indefinite Weight Function In R~N

As we all know,the eigenvalues and eigenfunctions of linear differential operators are one of the core of operator theory,and also are the basis for studying corresponding nonlinear problems.The dissertation mainly focuses on the spectrum theory of fractional Laplacian operators with indefinite weight function in RN.As an application of the the-ory,we further study the corresponding bifurcation problems.The main contents are as follows:Firstly,We give a survey to the research status of classical Laplacian operator and the fractional Laplacian operator's spectrum theory,and briefly introduce the main results of this paper.At the same time,we also introduce the definitions of fractional Laplacian operator,fractional Sobolev space and its related properties.Secondly,by using the variational method,the operator theory and the regularity theory,we study the eigenvalues and eigenfunctions of the eigenvalue problem(-?)su=?gu,x?RN,where 0<s<1,N>2s and the indefinite weight function g is bounded or unbounded.The existence of real eigenvalue sequence of the above problem are obtained when g satis-fies the appropriate conditions,especially,the principal eigenvalue ?1 is simple,isolated.The main results of spectrum theory partially improved the known results about Allegret-to[Proc.Amer.Math.Soc.,1992],Brown et al[Proc.Amer.Math.Soc.,1990;Proc.Royal Soc.Edinburgh,1993],Szulkin et al[Studia Mathematica,1999]and Servadei et al[Disc.Cont.Dyna.Syst.,2013].Lastly,we consider the corresponding bifurcation problems and obtain that(?1,0)is a bifurcation point of the bifurcation problem(-?)su = ?gu + f(x,?,u),x?RN under f and g satisfying certain conditions,where ?1 is the principal eigenvalue of the eigenvalue problem(-?)su = ?gu,x?RN.This result also partially generalizes the known results about Drabek et al[Trans.Amer.Math.Soc.,1997].