Positive Ground State Solutions for a Critical Nonlocal Problem in Dimension Three

In this paper, we are interested in the following nonlocal problem with critical exponent {- ( a - b integral(Omega) |del u|(2)dx) Delta u = lambda|u|(p-2) u +|u|(4)u, x is an element of Omega, u = 0, x is an element of partial derivative Omega, where a,b are positive constants, 2< p < 6, Omega is a smooth bounded domain in R-3 and lambda > 0 is a parameter. By variational methods, we prove that problem has a positive ground state solution u(b) for lambda > 0 sufficiently large. Moreover, we take b as a parameter and study the asymptotic behavior of u(b) when b SE arrow 0.