Existence Of Multiple Positive Solutions For Nonhomogeneous Fractional Laplace Problems With Critical Growth

In this thesis we discuss the existence of multiple solutions for nonhomogeneous frac-tional Laplace problems with critical growth.Namely,we consider the problem:#12 where s ?(0,1)and(-?)s is the fractional Laplace operate,?(?)RN(N>2s)is a smooth bounded domain,p=2s*:=2N/N-2s is the fractional Sobolev exponent,g?C0(?),g(x)?0(x??)and g(x)(?)0 in ?,??0,?? 0.This thesis first uses the method of monotonic iteration to prove that when ??[0,?1)(the eigenvalue ?1 is the first eigenvalue of the operator(-?)s under Dirichlet bound-ary conditions),there exists a positive constant ?*such that the problem(*)admits a positive minimal solution for all ??(0,?*]and admits no positive solution for ?>?*.Then this paper uses variational methods to prove that when ? E[0,?1),? E(0,?*),the problem(*)admits at least two positive solutions.