The Existence Of Positive Solutions For Kirchhoff Type Problem And Fractional Elliptic Problems In Exterior Domains

In this paper,we are concerned with the existence of positive ground state solutions for nonlinear critical Kirchhoff type problem and the existence of positive ground state and bound state solutions for a class of critical nonlocal problems of mixed fractional order Laplace operator in exterior domains by using the variational method.The main research content is stated as follows:The first part is devoted to studying the existence of the positive ground state solutions for Kirchhoff type problem where a>0,b>0,4<p<6,and potentials V(x)satisfying suitable assumptions.We estab-lish the existence of the positive ground state solutions by Nehari manifold and concentration-compactness principle.The second part is to study the existence of positive ground state and bound state solutions for critical nonlocal elliptic problem where ? is an exterior domain with RN\? bounded or ? is the whole RN,2<p<2s*,s ?(0,1),2s*=2N/N-2s(N>2s),?>0,and potentials V(x)satisfying suitable assumptions.We prove the existence of ground state solution by Nehari manifold and the global compact-ness Lemma,then the homotopy mapping is constructed by energy estimates and barycenter mapping,and the existence of bound state solution is proved by combining deformation lem-ma.