Positive ground state solutions for Choquard equations with lower critical exponent and steep well potential. (August 2021)

• Record Type: Journal Article
• Authors: Li, Yong-Yong; Li, Gui-Dong; Wu, Xing-Ping
• Is Part Of: Applied mathematics letters. Volume 118(2021)
• Publication Date: 2021-08
• Subjects: Choquard equation -- Lower critical exponent -- Steep well potential -- Ground state solution -- Concentration; Applied mathematics -- Periodicals; 519.05
• DOI: 10.1016/j.aml.2021.107151
• Languages: English
• ISSNs: 0893-9659

Abstract

Abstract: In this paper, we investigate the following non-autonomous Choquard equation − Δ u + λ V ( x ) − μ u = I α ∗ | u | 2 ∗ α | u | 2 ∗ α − 2 u in R N, where N ≥ 3, λ > 0, μ ∈ R, I α is a Riesz potential of order α ∈ ( 0, N ), 2 ∗ α = N + α N, the function V ∈ C ( R N, R ) is nonnegative and has a potential well Ω ≔ int V − 1 ( 0 ) such that − Δ possesses a sequence of Dirichlet eigenvalues 0 < μ 1 < μ 2 < ⋯ < μ n → + ∞ on Ω . Assume μ < μ 1, by the Nehari manifold method we prove the existence and concentration of positive ground state solutions for λ large enough.