Existence Of Solutions For Fractional Quasi-linear Elliptic Equations

In this thesis,by using critical point theory,the existence and multiplicity results of solutions for critical fractional p-Laplacian problems are obtained.The whole thesis is divided into two parts.In the first part,we study the asymmetric critical fractional p-Laplacian equation:(?) Where(-Δ)psu is the the fractional p-Laplacian of u,which is defined as:(?)∈RN,(?)Bε(x)={y∈RN:|x-y|<ε},N>ps，s∈(0,1),λ>0,ps*=Np|(N-sp) is the fractional critical Sobolev exponent,Ω(?)RN is an open bounded domain,and u+(x)= max{u(x),0}.With the help of the mountain pass lemma,we obtain the existence of the nontrivial solution for N = sp2 and 0<λ<λ1.By the linking theorem based on the Z2-cohomological index,we obtain the existence of the nontrivial solution for N>sp2 and λ are not the eigenvalues of(-Δ)psu.In the second part,we study Kirchhoff type equation involving the fractional p-Laplacian problems with critical growth:(?) where Ω(?)RN is an open bounded domain,the positive function M is non-degenerate and continuous,μ>0,N>ps,f(x,u)is the Caretheodory function that satisfies the subcritical condition.We obtain the existence of multiple solutions by the symmetric mountain pass lemma.