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

Abstract

We are concerned with the following nonlocal problem involving critical Sobolev exponent −a−b∫Ω∇u2dxΔu=λuq−2u+δu2u,x∈Ω,u=0,x∈∂Ω, where Ω is a smooth bounded domain in ℝ4, a,b>0, 1<q<2, δ, and λ 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 δ↘0. In the proofs, we apply variational methods.