Semiclassical States for Fractional Choquard Equations With a Potential Well
Funding: J. Yang is supported by the Natural Science Foundation of Hunan Province of China (2023JJ30482 and 2022JJ30463), the Research Foundation of Education Bureau of Hunan Province (23A0558 and 22A0540), and the Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province.
ABSTRACT
In this paper, we studied the existence and concentration behavior of positive solutions for a fractional Choquard equation involving a small parameter and a nonlocal nonlinearity. Under suitable assumptions on the potential function and the nonlinear term, we established two main results. First, for sufficiently small parameters, the equation admits multiple positive solutions, with the number of solutions depending on the topological structure of the potential's minimum set. Moreover, these solutions concentrate near this set as the parameter tends to zero. Second, we proved the existence of a single-peak solution whose peak converges to the potential's minimum set, and its rescaled version approaches a least energy solution of the associated limiting problem.
Conflicts of Interest
The authors declare no conflicts of interest.
Open Research
Data Availability Statement
The authors have nothing to report.