Existence And Multiplicity Of Solutions For Some Fractional Elliptic Problems

In this paper,we study existence and multiplicity of solutions for some kinds of fractional elliptic problems.Firstly,we consider the following problem(?)where ? is a smooth bounded set in Rn,M(t)=a+bt?-1,s ?(0,1)fixed,n>ps,1<q<p<?p?r?p?*,p?*=(n-?)p/n-ps is fractional critical Hardy-Sobolev exponent for 0 ??<ps<n,(-?)ps is fractional p-Laplacian operator.By fibering maps and Nehari manifold,and using concentration compactness and truncation technique to overcome the lack of compactness,we obtain at least two solutions to the problem for both Hardy-Sobolev subcritical and critical cases.Secondly,we consider the following of degenerate Kirchhoff problem(?)where ? is an open bounded set in Rn,p>1,??0,0<q<?+p-1<r<ps*-1 with ps*=-np/n-p for n>ps and s ?(0,1),f(x)and g(x)are sign-changing functions.By using Nehari manifold and fibering maps we obtain this problem at least two solutions.Thirdly,we study fractional p-Kirchhoff problem with critical Hardy-Sobolev system(?)where ? is a smooth bounded set in s?(0,1),1<q<p<?p<?+??ps*(k),?,?>1,ps*(k)=(n-k)p/n-ps is Hardy-Sobolev critical exponent for 0 ?k<ps<n.f(x)?0 and g(x)>0 are countinuous functions in ?.By fibering maps and Nehari manifold,and using truncation technique and Brezis-Lieb lemma to overcome the lack of compactness,we obtain the above system at least two solutions.Finally,we consider multiplicity of solutions for fractional p&q-Laplacian system involving critical concave-convex nonlinearities(?)where ? is a smooth bounded set in Rn,0<s<1,1<r<q<p<n/s,and ?>1,?>1 satisfy ?+?=ps*with ps*=np/n-ps being the fractional critical Sobolev exponent.By using Lusternik-Schnirelmann's theory,we obtain existence of multiple solutions for fractional p&q-Laplacian problems.