Solutions Of Fractional P-laplacian Problems With Indefinite Weight Functions

The dissertation mainly focuses on the existence and multiplicity of solutions for fractional p-Laplacian problems with indefinite weight functions#12 where s ∈(0,1),λ>0,n>ps,p≥2,Ω(?)Rn is an open bounded subset with smooth boundary,a,b:Ω→R are sign-changing continuous functions.The nonlocal operator(-△)ps is defined as follows:#12 where P.V.means cauchy principal value,If p=2,then(-△)ps=(-△)s.In both 1<β<p and p<β<p*cases,p*=np/n-pα is the fractional critical Sobolev exponent.When λ<λ1,we get the existence of single solution using both the fibering map and Nehari manifold in these situations,where λ1 denotes the first eigenvalue of the following eigenvalue problem#12 If ∫Ωb(x)φ1βdx<0,φ1 denotes the nonnegative eigenfunction corresponding to λ1 we prove the problem has at least two nonnegative solutions whenever λ1<λ<λ1+δ.Ifλ=λ1,∫Ωb(x)φ1βdx<0,p<β<p*we prove the problem has a nonnegative solution.And we also study the limit behavior for these solutions in these situations.