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),?>0,n>ps,p?2,?(?)Rn is an open bounded subset with smooth boundary,a,b:??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<?<p and p<?<p*cases,p*=np/n-p? is the fractional critical Sobolev exponent.When ?<?1,we get the existence of single solution using both the fibering map and Nehari manifold in these situations,where ?1 denotes the first eigenvalue of the following eigenvalue problem#12 If ??b(x)?1?dx<0,?1 denotes the nonnegative eigenfunction corresponding to ?1 we prove the problem has at least two nonnegative solutions whenever ?1<?<?1+?.If?=?1,??b(x)?1?dx<0,p<?<p*we prove the problem has a nonnegative solution.And we also study the limit behavior for these solutions in these situations.