Existence Of Solutions For Fractional Elliptic Systems

Nonlinear elliptic equations and systems are derived from mathematical models in natural sciences and engineering.Recently,it is more significant to study elliptic systems involving fractional operators which can be used to solve problems globally.So the existence and multiplicity of solutions for two kinds of systems are investigated in this thesis which can be divided into two parts:In the first part,the fractional Brezis-Nirenberg system involving critical Choquard nonlinearities are considered.Under appropriate conditions,existence of solutions to the system is obtained,which is a great extension of classical results of Brezis and Nirenberg for the local case to the problem with Choquard nonlinearity.When the nonlinearity is subcritical,we get a positive solution for this problem by minimizing sequences.When the nonlinearity is critical,we achieve a nonnegative solution by the Mountain Pass theorem.When the nonlinearity interacts with the fractional Laplacian spectrum,the existence of a nontrivial solution is established by the Linking theorem.In the second part,we investigate the fractional p-Laplacian system involving critical Hardy-Sobolev exponents.The existence and multiplicity of solutions to this system are obtained by the Nehari manifold techniques under certain conditions of the pair of the parameters(?,9).For the critical case,compact conditions hold true in limited level sets.When constructing estimates of the critical value,the major difficulty is the lack of an explicit formula for a minimizer in fractional Hardy-Sobolev inequality.We get around this difficulty by working with certain asymptotic estimates for the minimizers.