Concentrated solutions to fractional Kirchhoff equations with inhomogeneous perturbation

This paper is concerned with normalized solutions for a class of fractional Kirchhoff equations with an inhomogeneous perturbation in R 2 . We study the constrained minimization problem associated with the corresponding nonlocal energy functional under a prescribed L 2 -mass. The interaction between the Kirchhoff term, the fractional Laplacian and the spatially dependent coefficient f ( x ) leads to several compactness and asymptotic difficulties. For the zero-potential case, we establish existence and nonexistence results for constrained minimizers. When a trapping potential is present, we prove the existence of minimizers in the subcritical case and characterize the threshold in the critical case p = 4 s . In the mass-critical case p = 2 s , we analyze the concentration behavior of non-negative minimizers as the Kirchhoff parameter tends to zero. We prove that the minimizers concentrate at global minimum points of the potential and, after rescaling, converge to the unique positive radial solution of the limiting fractional scalar field equation. A sharper description of the concentration location and the energy asymptotics is obtained when the potential is almost homogeneous near its minimum points.