Multiple Positive Solutions and Estimates of Extremal Values for a Nonlocal Problem with Critical Sobolev Exponent and Concave-Convex Nonlinearities

We are concerned with the following nonlocal problem involving critical Sobolev exponent {-(a-b integral(omega)|& nabla;u|(2)dx)delta u=lambda|u|(q-2)u+delta|u|(2)u, x epsilon omega, u=0, x epsilon & part;omega, where omega is a smooth bounded domain in R-4, a,b > 0, 1 < q < 2,delta, and lambda are positive parameters. We prove the existence of two positive solutions and obtain uniform estimates of extremal values for the problem. Moreover, the blow-up and the asymptotic behavior of these solutions are also discussed when b?0 and delta?0. In the proofs, we apply variational methods.