Existence And Multiplicity Of Solutions Of Fractional Elliptic Equations

This paper studies existence and multiplicity of solutions for fractional elliptic equation via variational methods, critical point theory and implicit function theory.Firstly,we study the non-local fractional elliptic equation wherenW ìR is an open bounded domain with smooth boundary, sn >2, s?(0,1) is fixed,sD-)( is the fractional Laplace operator,which may be defined as when the nonlinear term f satisfying the following conditions(f1)f: W ′R ®R is a Carathéodory function, there exist C>0 and q?)2,1(,that1(,)(1)qf x u C u-￡ +;(f2) lim(,)t F x u®￥= +￥,uniformly for xW?, where0(,)(,)uF x u = òf x s ds The fractional elliptic equations are studied,by using the Mountain-Pass theorem,Saddle Point Theorem and Ekeland’s variational Principle in critical point theory.That is, there exists 00e>, the fractional elliptic equation has at least three solutions for all),(101-?lell, where1l is the principal eigenvalue.And then, we concerned the fractional elliptic equation wherenW ìR is an open bounded domain with smooth boundary, sn >2, s?(0,1) is fixed and f is a continuous function on ′WR,sD-)( is the fractional Laplace operator,which may be defined as when the nonlinear term f satisfying the following conditions (f1) f(x,-u) = -f(x,u) for all xW? and Ru ?;(f2)there exist constants 01a> and sn snr221-+<< such that(f3) ux F0),( 3, for all),(Rux ′W? and ￥=￥®2limu u, uniformly for x?W,where0(,)(,)uF x u = òf x s ds;(f4) there exists a constant M>0 such that M u xfuux Fu u<+-￥® 1),(2),(suplim2, uniformly for x?W.We obtain existence of infinite solutions for fractional elliptic equation via a variant of fountain theorem.