In this paper, we study the existence, non-existence and asymptotic behavior of positive ground states for the nonlinear Choquard equation: (0.1) β Ξ β’ π’ + π β’ π’ = ( πΌ πΌ β πΉ β‘ ( π’ ) ) β’ πΉ β² β‘ ( π’ ) , π’ β π» 1 β‘ ( β π ) , where πΉ β‘ ( π’ ) = | π’ | π + πΌ π + πΊ β‘ ( π’ ) with πΊ β‘ ( π’ ) = β« π’ 0 π β‘ ( π ) π π , π β₯ 3 is an integer, πΌ πΌ is the Riesz potential of order πΌ β ( 0 , π ) and π > 0 is a frequency parameter. Under some mild subcritical growth assumptions on π β πΆ β‘ ( [ 0 , β ) , [ 0 , β ) ) , we establish a sharp threshold result for the existence of ground states, and an asymptotic characterization of the ground state solutions as π β 0 . In particular, if π β‘ ( π ) βΌ π π β 1 as π β 0 for some π β ( π + πΌ π , π + πΌ π β 2 ) , then if π < π + πΌ + 4 π , (0.1) admits a ground state for all π > 0 , and if π β₯ π + πΌ + 4 π , then a threshold phenomena occur: there exists π π > 0 such that (0.1) has no ground state for π β ( 0 , π π ) and admits a ground state for π > π π . If π β‘ ( π ) β π β’ π π β 1 as π β 0 for some π > 0 and π β ( π + πΌ π , m i n β‘ { π + πΌ π β 2 , π + πΌ + 4 π } ) , we show that as π β 0 , the ground state solutions of (0.1) , after a suitable rescaling, converges in π» 1 β‘ ( β π ) to a particular solution of the Hardy-Littlewood-Sobolev critical equation π’ = π + πΌ π β’ ( πΌ πΌ β | π’ | π + πΌ π ) β’ | π’ | π + πΌ π β 2 β’ π’ . It turns out that the limit profiles are determined solely by the locations of ( π , π ) in ( 0 , + β ) Γ ( π + πΌ π , m i n β‘ { π + πΌ π β 2 , π + πΌ + 4 π } ) . We also establish a novel sharp asymptotic characterization of such a rescaling.
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Ma et al. (2026) studied this question.
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