Year: 2025
Type: article
Abstract: This paper focuses on static solutions for the following Choquard equation with zero mass and Coulomb potential $$\begin{aligned} -\Delta u+\left( \frac{1}{4\pi |x|}*u^2\right) u=\mu |u|^{p-2}u+(I_{\alpha }*|u|^{\alpha +3})|u|^{\alpha +1}u, \ \ \ \ x\in {\mathbb {R}}^{3}, \end{aligned}$$ - Δ u + 1 4 π | x | ∗ u 2 u = μ | u | p - 2 u + ( I α ∗ | u | α + 3 ) | u | α + 1 u , x ∈ R 3 , where $$\mu >0$$ μ > 0 , $$\frac{18}{7} 18 7 < p ≤ 6 , $$\alpha \in (0, 3)$$ α ∈ ( 0 , 3 ) , $$\alpha +3$$ α + 3 is the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality, $$I_{\alpha }{:}\,\mathbb {R}^3\rightarrow \mathbb {R}$$ I α : R 3 → R is the Riesz potential, and $$\frac{1}{4\pi |x|}$$ 1 4 π | x | is the Coulomb potential. By carefully analyzing the intricate interplay between the power and Coulomb terms, we establish three types of variational geometries of the problem and prove the following existence results based on the behavior of p: the existence of two solutions, one being a local minimizer and the other of mountain-pass type, for an explicit range $$0<\mu <\mathrm {Const.}$$ 0 < μ < Const
Source: Mathematische Annalen
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W4409309903