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 ∇ u 2 d x Δ u = λ u q − 2 u + δ u 2 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where Ω is a smooth bounded domain in ℝ 4 , 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.