Existence thresholds and limit profiles of ground states for lower critical Choquard equations with general nonlinearities

Abstract

In this paper, we study the existence, non-existence and asymptotic behavior of positive ground states for the nonlinear Choquard equation: (0.1) − Δ ⁢ 𝑢 + 𝜀 ⁢ 𝑢 = ( 𝐼 𝛼 ⁎ 𝐹 ⁡ ( 𝑢 ) ) ⁢ 𝐹 ′ ⁡ ( 𝑢 ) , 𝑢 ∈ 𝐻 1 ⁡ ( ℝ 𝑁 ) , where 𝐹 ⁡ ( 𝑢 ) = | 𝑢 | 𝑁 + 𝛼 𝑁 + 𝐺 ⁡ ( 𝑢 ) with 𝐺 ⁡ ( 𝑢 ) = ∫ 𝑢 0 𝑔 ⁡ ( 𝑠 ) 𝑑 𝑠 , 𝑁 ≥ 3 is an integer, 𝐼 𝛼 is the Riesz potential of order 𝛼 ∈ ( 0 , 𝑁 ) and 𝜀 > 0 is a frequency parameter. Under some mild subcritical growth assumptions on 𝑔 ∈ 𝐶 ⁡ ( [ 0 , ∞ ) , [ 0 , ∞ ) ) , we establish a sharp threshold result for the existence of ground states, and an asymptotic characterization of the ground state solutions as 𝜀 → 0 . In particular, if 𝑔 ⁡ ( 𝑠 ) ∼ 𝑠 𝑞 − 1 as 𝑠 → 0 for some 𝑞 ∈ ( 𝑁 + 𝛼 𝑁 , 𝑁 + 𝛼 𝑁 − 2 ) , then if 𝑞 < 𝑁 + 𝛼 + 4 𝑁 , (0.1) admits a ground state for all 𝜀 > 0 , and if 𝑞 ≥ 𝑁 + 𝛼 + 4 𝑁 , then a threshold phenomena occur: there exists 𝜀 𝑞 > 0 such that (0.1) has no ground state for 𝜀 ∈ ( 0 , 𝜀 𝑞 ) and admits a ground state for 𝜀 > 𝜀 𝑞 . If 𝑔 ⁡ ( 𝑠 ) ≃ 𝑎 ⁢ 𝑠 𝑞 − 1 as 𝑠 → 0 for some 𝑎 > 0 and 𝑞 ∈ ( 𝑁 + 𝛼 𝑁 , m i n ⁡ { 𝑁 + 𝛼 𝑁 − 2 , 𝑁 + 𝛼 + 4 𝑁 } ) , we show that as 𝜀 → 0 , the ground state solutions of (0.1) , after a suitable rescaling, converges in 𝐻 1 ⁡ ( ℝ 𝑁 ) to a particular solution of the Hardy-Littlewood-Sobolev critical equation 𝑢 = 𝑁 + 𝛼 𝑁 ⁢ ( 𝐼 𝛼 ⁎ | 𝑢 | 𝑁 + 𝛼 𝑁 ) ⁢ | 𝑢 | 𝑁 + 𝛼 𝑁 − 2 ⁢ 𝑢 . It turns out that the limit profiles are determined solely by the locations of ( 𝑎 , 𝑞 ) in ( 0 , + ∞ ) × ( 𝑁 + 𝛼 𝑁 , m i n ⁡ { 𝑁 + 𝛼 𝑁 − 2 , 𝑁 + 𝛼 + 4 𝑁 } ) . We also establish a novel sharp asymptotic characterization of such a rescaling.