2025/02/25 by S.H. Mo, Shiwang Ma, Mo, Shuai +1
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #FOS: Mathematics #Geometry and complex manifolds #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2502.18254
openalex publication_date 2025/02/25 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28
We study the normalized solutions to the following Choquard equation \aligned amp;-Δu + λu =μg(u) + γ(Iα* |u|(N+α)/(N))|u|(N+α)/(N)-2u amp; in ℝN \endaligned under the L2-norm constraint ‖u‖2=c. Here γ>0, N≥ 1, α∈(0,N), Iα is the Riesz potential, and the unknown λ appears as a Lagrange multiplier. In a mass supercritical setting on g, we find regions in the (c,μ)--parameter space such that the corresponding equation admits a positive radial ground state solution. To overcome the lack of compactness resulting from the nonlocal term, we present a novel compactness lemma and some prior energy estimate. These results are even new for the power type nonlinearity g(u)= |u|q-2u with 2+(4)/(N)0. Based on some analytical ideas the limit behaviors of the normalized solutions, we verify some threshold regions of η such that the corresponding equation has no positive least action solution or admits multiple positive solutions. To the best of our knowledge, this seems to be the first result concerning the non-existence and multiplicity of positive solutions to Choquard type equations involving the lower critical exponent.