Multiple Positive Solutions and Estimates of Extremal Values for a Nonlocal Problem with Critical Sobolev Exponent and Concave-Convex Nonlinearities. (2nd June 2022)

• Record Type: Journal Article
• Authors: Shi, Zhigao; Qian, Xiaotao
• Other Names: Secchi Simone Academic Editor.
• Is Part Of: Journal of function spaces. Volume 2022(2022)
• Publication Date: 2022-06-02
• Subjects: Function spaces -- Periodicals; 515.7305
• DOI: 10.1155/2022/1011342
• Languages: English
• ISSNs: 2314-8896

Abstract

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.