LAUSR

In this study, we explore both the nonexistence and existence of a nonzero solution for the following Choquard equation (CE): −∆u+𝒱(ϰ)u=−∆D0−α2*|u|αN+1|u|αN−1u,inℝN,u∈H1ℝN, where α∈(0,N),N≥3,$$ \alpha \in \left(0,N\right),N\ge 3, $$ * represents the convolution within ℝN$$ {\mathbb{R}}^N $$ , 𝒱(ϰ)∈L∞ℝN, the exponent αN+1$$ \frac{\alpha }{N}+1 $$ denotes the critical value with respect to the lower Hardy–Littlewood–Sobolev inequality (HLSI), and −∆D0−α2$$ {\left(-{\Delta }^{{\mathfrak{D}}_0}\right)}^{\frac{-\boldsymbol{\alpha}}{\mathbf{2}}} $$ represents the Riesz fractional Laplacian with zero boundary condition. The Riesz potential within this context has not been previously investigated. Altering the Riesz potential in the Choquard equation is significant as it affects the properties and solutions of the equation. Practically, this change is important due to its applications in various fields of physics and mathematical analysis. We establish the existence of a ground state solution (GSS) in ℝN$$ {\mathbb{R}}^N $$ . This analysis relies on the application variational techniques.