Multiplicity and concentration results for fractional Choquard equations: Doubly critical case. (September 2020) - Su, Yu; Wang, Li; Chen, Haibo; Liu, Senli
Abstract: In this paper, we consider the following fractional Choquard equation: ε 2 s ( − Δ ) s u + V ( x ) u = ε − α ( I α ∗ F ( u ) ) F ′ ( u ), x ∈ R N, where N ⩾ 3, s ∈ ( 0, 1 ] and α ∈ ( 0, N ), ε > 0 is a parameter, I α is the Riesz potential, and F ( u ) ≔ 1 2 α ♯ | u | 2 α ♯ + 1 2 α ∗ | u | 2 α ∗, where 2 α ♯ = N + α N and 2 α ∗ = N + α N − 2 s are lower and upper critical exponents in the sense of the Hardy–Littlewood–Sobolev inequality. Firstly, by the refined Sobolev inequality with Morrey norm, we show a generalization of Lions type theorem. Secondly, combining this theorem with variational methods, we show the multiplicity and concentration of positive solutions for above equation. Moreover, the multiplicity and concentration results are obtained in the case where F ( u ) has the lower critical exponent 2 α ♯, which remains unsolved in the existing literature.