Existence And Multiplicity Of Solutions For Fractional P-Laplacian Equations With Critical Sobolev Exponents

In this thesis,we mainly study the existence and multiplicity of solutions for the following fractional p-Laplacian problem:(?)where ?(?)Rn is a bounded smooth domain,p>1,s ?(0,1),n>sp,(-?)ps is fractional p-Laplacian operator,? is a non-negative real parameter,q ?[p,ps*)and ps*=-np/n-sp is fractional Sobolev exponent.We obtain that the problem has no nontrivial solution when ?=0.When p>2 and ? is positive and sufficiently small,if q=p,n>sp2 or q?(p,ps*),n>+2s(pq-p-q)/pq 2p p2 q,using the Ekeland variational principle,we deduce the problem has at least a nonnegative nontrivial solution.In addition,Using the Lusternik-Schnirelman theory,we show the problem has at least cat?(?)solutions.These results generalizes the main results about Alves et al[JMAA,2003]and Figueiredo et al[CVPDE,2018].